Forget trying to convince a certain section of the fandom of this. They have established the narrative among themselves that "Picard is a soulless robot now" and nothing's gonna get them to realize that he doesn't look like Data on the inside and that he is flesh and blood and that therefore a doctor and not an engineer is still the correct person to take care of him in case of injury. (Maybe not with an old-fashioned - by 24th/25th century standards - defibrillator tho, granted.)
"Picard is a soulless robot now" is the same stuff as "Disco has been canceled", "Kurtzman has been fired", etc etc - all excellent examples of the "tell a lie often enough and it becomes the truth". Or rather, an alternative Trek fact in certain fandom sub sections.
Yeah, they could chop his arm off, veins everywhere, blood spewing, and they'll be like "Nope! Still an android!"
You'd think they'd never seen Battlestar Galactica and didn't know about the Final Five. Although, with this crowd, maybe they haven't.
This confusion is the result of bad writing.
It is not the result of antagonism towards NuTrek or trolling from the fans. Stop blaming fans. The writers were not able to make their point clear. It just shows how bad the storytelling in Picard S1 was.
Some people don't like it that NuTrek critics use the phrase "bad writing", so let me elaborate. Let me explain in detail why the confusion about robot Picard is "bad writing":
As far as I remember (I have watched Picard only once, over two years ago), there are only two scenes in the entirety of season one where the biological nature of the synthetics is addressed (and the first scene is not unambiguous if you are a TNG fan).
1) In a very short dialogue between Picard and Agnes at the end of S01E01
2) Another short scene in S01E10 where you could see a ripped-out eye of a synth and it was biological.
In the scene in S01E01 between Picard and Agnes, Picard talks about a "synthetic that looks fully human".
"Looks" fully human, not "is" fully human.
Not only is the language the writers are using confusing (Picard also uses the phrase "'android' out of flesh and blood". more language confusion), but more importantly, anybody who knows TNG will automatically think of Data's mother.
Data's mother is a synthetic that looks fully human.
There was an entire TNG episode about a Soong-type android that looks fully human. No wonder fans think that Picard is a style of Soong-type android like Data's mother.
In a single TNG episode ("Inheritance"), we spent more time seeing the insides of Data's mother than we see the biological nature of the synthetics in an entire season.
Think about that in terms of "mindshare". How much "mindshare" does Data's mother have among fans, compared to the new biological synthetics? A good writer should consider that. The writers simply did not consider how much mindshare Data's mother has among fans.
The nature of the synthetics:
The biological synthetics are not fully biological. The biological synthetics also have positronic brains and superhuman strength.
It doesn't help your case to establish that the biological synthetics are mostly biological when characters constantly talk about "positronic brains" and "fractal neuronic cloning", and show that the bio synthetics are super fighters and tear through metal bulkheads with their bare hands.
More visual and language confusion:
The Coppelius androids are all bio synthetics, but they are called "androids".
Almost all Coppelius androids (except Soji and Dahj) have golden skin. The Mars androids have golden skin.
But all Coppelius androids are bio synthetics and the Mars androids are mechanical.
Let me go back to the scene in S01E01 between Picard and Agnes and dissect it even more and give you a good example of "bad writing":
In this scene, Picard and Agnes constantly switch about what kind of synthetics they talk about. You have to pay very close attention to not get confused.
First Picard asks about biological synthetics ("out of flesh and blood").
Then Agnes switches the topic and starts talking about classic mechanical Soong-type androids. She talks about the androids that attacked Mars and that they build here.
Then Picard switches topic again and asks about biological synthetics, BUT in a way that if you are a TNG fan you think that he asks about Data's mother style mechanical Soong-type androids ("looks fully human").
Then Agnes switches again about what kind of androids they talk about. She talks about B4 and positronic brains.
Do you see how this adds to the confusion? In the minds of the writers, they let the characters talk about biological synthetics, in the minds of TNG fans the characters talk about Data's mother-style mechanical Soong-type androids.
No wonder people are confused. The writers did a poor job getting their point across. There was not a single unambiguous scene (for TNG fans who have Data's mother on their mind) where characters talk about the biological nature of these Androids and made it unambiguously clear that these are not Data's mother type androids.