I'd much rather their reason for not repeating a mission having to do with concerns about a paradox. The whole "the same matter can't occupy the same space" nonsense (which is what they were implying here without actually saying it) is just so much, well, nonsense. At no point whatsoever would the same matter be occupying the same space even if you shook hands with temporal duplicate. I mean, that's like saying that if you, right now, were to touch your index finger to your other index finger, the universe would spontaneously explode for the exact same (and very stupid) reason.
If you don't want to have an easy out with time travel like that, just come up with something that actually makes some amount of sense. A devastating temporal ripple that will tear about your present/future and put the entire project at risk. The paradox winking you out of existence, thereby destroying every bit of work you've done or will do. Impossible to "cross the streams" created by the time travel device. Wormholes collapsing on themselves if they intersect, sending you hurling into netherspace for all eternity. Whatever. Just anything other than the same-space malarkey. Please!
Moreover, if you were to go back in time and shake hands with yourself, unless that time jump was fairly short it wouldn't be the "same matter" since the matter of our bodies is constantly being destroyed/dying and being replaced. It takes your skin a little over a month to entirely replace itself. So after a month touching yourself wouldn't be "the same matter."
I too hope they go with something closer to it just creating too much of a paradox. As for sending another team back, it seems they threw this team together pretty quickly, maybe its a matter of there's no one else to send? Why let more people know about this secret project? (The more people who know the more likely it is for it to be leaked out.) And then just plain not wanting to send back more people and potentially make things worse. What then? Send back even *more*? It's probably best to not pile-on and just deal with the consequences. Besides, from the project leaders' point of view nothing has changed.
Which, again, sort of brings up the usual time-travel discrepancy of how can the past be changed to begin with to give the characters a chance to do something about it, and realize the change while
at the same time the past be changed and for characters to
not be aware of it. It's problems like these that makes time-travel stories so hard to tell because you easily fall into holes like this. You need both the "ticking clock" and "repercussions for failure."
And maybe I missed some expositional dialogue by why send them back so close to the time the other guy went back? They say they don't know where the guy went but,
when so..... Doesn't this mean they have to go back quite a good deal of time in order to find him? It's not clear, to me, if the time machine moves spatially (beyond needing to move itself with the cosmic movement of the universe and all of its components over time; we'll say what ever connection between points in time the machine uses to travel moves with everything) as well as temporally, but if it can doesn't that make it pretty damn hard to know what he's going back to change if he has the whole planet (or in this case all of America since he's specifically targeting American history for whatever reason) as well as all of history to choose from? And if it can't move spatially seems to me that limits the things he can go back and change without having to travel on arrival, still making it hard to find him.
I'd accept it if they just said they had a *whatever* Database that can compare "original recorded history" with "temporally local history" to give them a destination in all 4 dimensions, it's just a matter of getting to him and stopping him because neither is 100% accurate; they can get to within a range before but not the exact moment. Because knowing exactly when is sort of convenient. And, I dunno, aren't these people pretty much always destined to fail while at the same time
he is destined to fail? For the story to continue
they can never capture him because then the series would be over. At the same time
he can never fully succeed because then the series would be over, he accomplished his goal and has no need for further meddling and presumably his changes were disastrous enough to prevent any attempts at stopping him from occurring; meaning the dis-invention of time travel. Which is a paradox because if time-travel is never invented then he can't go back in time to
prevent it from being invented and...
Oh dear, now I've gone cross-eyed.
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The prototype was just that: a prototype. And the prototype of any invention is always going to look a lot rougher and less refined than the first iteration of it, or a "production line" model. (Thanks, Knight Rider's KARR for that way of thinking.

) When something is invented it's not like it comes out perfect in function and appearance the first time around. Hell, the DeLorean was a "prototype" of sorts as it had exposed wiring and other components that could be damaged just from ordinary use of the vehicle and as well as an inefficiency considering all of the switches and buttons it had on the overhead panel and other areas inside the cabin. Had Doc had the will, time, and money after the DeLorean was successful he likely would have bought a second DeLorean and built a final version of the time-machine that was refined, the wiring under the body panels or shrouded, refined circuitry and electronics (possibly using future technology) to prevent the need for all of the switches and such inside the dash. A better-built and less prone to failure time read-out display, etc. He already improved the power source on it.
On the "can't go back to meet yourself" thing. I don't think its strictly a "don't cross the streams" sort of thing where its mentioned purely for it to be broken, like the underwire in the bra was mentioned specifically so we'd know she had it for it to be used to pick the lock, but I don't think it was mentioned purely as a clumsy way to give themselves an excuse to not have infinite chances at trying to "solve a case", as it were.
I do think it is a "rule" that will be broken at some point down the line. If the series takes off and lasts for several seasons it's a rule they're going to break sooner or later if they don't already have one planned with her meeting a future self or something and the "the guy didn't come back in one piece" thing was just told to the team to discourage meeting themselves. They just don't do multiple missions because it just makes a complicated, dangerous, situation even more complicated and dangerous.