Whether you believe that Starfleet is a military, paramilitary or something else, there is a good reason that this topic warrants its own thread.
STARFLEET'S VIEW ON ITS SCIENCES DIVISION
There is a claim that Starfleet's primary purposes are science and exploration but I've found a prime example that directly contradicts it when you look at how Starfleet views its own sciences division.
TNG: "Tapestry"
[Bridge]
[Doctor's office]
[Ten Forward]
This example clearly demonstrates how Starfleet views its divisions:
Someone might say that it's just Picard, not all of Starfleet, but remember that he is an exemplary representative of Starfleet, a "hero" of the show.
STAR TREK SYMBOL
Starfleet has multiple divisions but Star Trek has always been represented by the Starfleet Command/Tactical symbol on screen:
What does that tell you about Starfleet's primary purpose?
EDIT 1: If it were only about the pips, Picard would have stayed in the command division but with a lower rank. Clearly, the episode is making a point about Starfleet divisions as well.
EDIT 2: The bottom line is this: Picard could have stayed in the command division as a lower-ranking conn officer (flight controller), whom he used to be in his youth. Flight controllers in TNG wore red uniforms and were part of the command division. However, the episode made it a point to change Picard's division to sciences. That's not reading "too deep" into it, that's a direct message of Starfleet's view on its divisions.
EDIT 3: Keep in mind that, at the time, Picard is in command of a flagship, so he's expected to reflect the majority view of Starfleet or he would not be commanding a flagship. He was also meant to be the star of the show and someone the audience was expected to root for. In other words, he was Starfleet's mouthpiece or spokesperson in TNG.
STARFLEET'S VIEW ON ITS SCIENCES DIVISION
There is a claim that Starfleet's primary purposes are science and exploration but I've found a prime example that directly contradicts it when you look at how Starfleet views its own sciences division.
TNG: "Tapestry"

[Bridge]
(Picard is a science lieutenant)
WORF: Can I help you, Mister Picard?
PICARD: Mister Worf.
(Worf takes a PADD from Picard)
WORF: This is not for me. You should take it to Commander La Forge in Engineering.
PICARD: What's happened?
WORF: Is something wrong?
PICARD: I'm not sure. Mister Worf, what is my rank and position?
WORF: You are a Lieutenant junior grade, Assistant Astrophysics officer.
DATA: Are you feeling all right?
PICARD: Who's the captain of this ship?
DATA: Captain Thomas Halloway. Perhaps I should escort you to Sickbay.
PICARD: No. I can find my own way there. Thank you, Commander.
WORF: Can I help you, Mister Picard?
PICARD: Mister Worf.
(Worf takes a PADD from Picard)
WORF: This is not for me. You should take it to Commander La Forge in Engineering.
PICARD: What's happened?
WORF: Is something wrong?
PICARD: I'm not sure. Mister Worf, what is my rank and position?
WORF: You are a Lieutenant junior grade, Assistant Astrophysics officer.
DATA: Are you feeling all right?
PICARD: Who's the captain of this ship?
DATA: Captain Thomas Halloway. Perhaps I should escort you to Sickbay.
PICARD: No. I can find my own way there. Thank you, Commander.

[Doctor's office]
PICARD: Q, what have you done?
Q: I've done exactly as I promised. I've returned you to the present.
PICARD: But this is not the present I remember. You said nothing would change.
Q: Nothing has changed, Jean-Luc, except for you. But then again, that's what you wanted, wasn't it? To change the man you were in your youth? Well, you did it. This is the man you are today. And You should be happy. You have a real heart beating in your chest, and you get to live out the rest of your life in safety, running tests, making analyses, and carrying reports to your superiors.
Q: I've done exactly as I promised. I've returned you to the present.
PICARD: But this is not the present I remember. You said nothing would change.
Q: Nothing has changed, Jean-Luc, except for you. But then again, that's what you wanted, wasn't it? To change the man you were in your youth? Well, you did it. This is the man you are today. And You should be happy. You have a real heart beating in your chest, and you get to live out the rest of your life in safety, running tests, making analyses, and carrying reports to your superiors.

[Ten Forward]
(Riker and Troi are at a table)
PICARD: [. . .] First of all, and I would like you to be absolutely straightforward with me. How would you evaluate me as an officer?
TROI: Well, er, your performance records have always been good. You're thorough, dedicated.
RIKER: Reliable, steady, (searching desperately) punctual.
PICARD: I see. What would you say if I told you that I believed that I was capable of being very much more.
RIKER: Perhaps we should discuss this at your next evaluation.
PICARD: I would appreciate it if we could discuss it now. You see, I feel that I would like to move beyond astrophysics to Engineering or Security, something that might even lead to Command.
RIKER: Frankly, Lieutenant, I don't think that's realistic.
PICARD: Why?
TROI: I really don't think this is the place to be discussing this.
PICARD: Please. This is important to me. I believe that I can do more.
TROI: Hasn't that been the problem all along? Throughout your career you've had lofty goals, but you've never been willing to do what's necessary to attain them.
PICARD: Would that be your evaluation as well, Commander?
RIKER: I think I have to agree with the Counsellor. If you want to get ahead, you have to take chances, stand out in a crowd, get noticed.
PICARD: I see.
RIKER: Now, we don't want to lose you. You're a very good officer.
PICARD: Just not one who stands out.
RIKER: Why don't I talk to Commander La Forge in Engineering and we'll see what we can do.
PICARD: But, Command?
RIKER: Well, we'll see.
[. . .]
(Riker and Troi leave)
PICARD: All right, Q, that's enough. You've made your point. Q?
PICARD: [. . .] First of all, and I would like you to be absolutely straightforward with me. How would you evaluate me as an officer?
TROI: Well, er, your performance records have always been good. You're thorough, dedicated.
RIKER: Reliable, steady, (searching desperately) punctual.
PICARD: I see. What would you say if I told you that I believed that I was capable of being very much more.
RIKER: Perhaps we should discuss this at your next evaluation.
PICARD: I would appreciate it if we could discuss it now. You see, I feel that I would like to move beyond astrophysics to Engineering or Security, something that might even lead to Command.
RIKER: Frankly, Lieutenant, I don't think that's realistic.
PICARD: Why?
TROI: I really don't think this is the place to be discussing this.
PICARD: Please. This is important to me. I believe that I can do more.
TROI: Hasn't that been the problem all along? Throughout your career you've had lofty goals, but you've never been willing to do what's necessary to attain them.
PICARD: Would that be your evaluation as well, Commander?
RIKER: I think I have to agree with the Counsellor. If you want to get ahead, you have to take chances, stand out in a crowd, get noticed.
PICARD: I see.
RIKER: Now, we don't want to lose you. You're a very good officer.
PICARD: Just not one who stands out.
RIKER: Why don't I talk to Commander La Forge in Engineering and we'll see what we can do.
PICARD: But, Command?
RIKER: Well, we'll see.
[. . .]
(Riker and Troi leave)
PICARD: All right, Q, that's enough. You've made your point. Q?

PICARD: [. . .] Are you having a good laugh now, Q? Does it amuse you to think of me living out the rest of my life as a dreary man in a tedious job?
This example clearly demonstrates how Starfleet views its divisions:
- Command (includes Starfleet Tactical): most respected, "lofty goals" career.
- Engineering and Security: fairly respected (middle of the road).
- Sciences: least respected, "dreary" and "tedious" job.
Someone might say that it's just Picard, not all of Starfleet, but remember that he is an exemplary representative of Starfleet, a "hero" of the show.
STAR TREK SYMBOL
Starfleet has multiple divisions but Star Trek has always been represented by the Starfleet Command/Tactical symbol on screen:

What does that tell you about Starfleet's primary purpose?
EDIT 1: If it were only about the pips, Picard would have stayed in the command division but with a lower rank. Clearly, the episode is making a point about Starfleet divisions as well.
EDIT 2: The bottom line is this: Picard could have stayed in the command division as a lower-ranking conn officer (flight controller), whom he used to be in his youth. Flight controllers in TNG wore red uniforms and were part of the command division. However, the episode made it a point to change Picard's division to sciences. That's not reading "too deep" into it, that's a direct message of Starfleet's view on its divisions.
EDIT 3: Keep in mind that, at the time, Picard is in command of a flagship, so he's expected to reflect the majority view of Starfleet or he would not be commanding a flagship. He was also meant to be the star of the show and someone the audience was expected to root for. In other words, he was Starfleet's mouthpiece or spokesperson in TNG.
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