Here's the perspective of an HSM mod.
The problem is that, being HSM still in beta state, and the questions related to mathematics, there's often resilience about it.
This is absolutely understandable, and I've seen a lot of opinions along these lines across SE regarding any meta site. That said, migrated questions to HSM have been extremely successful. We've rejected only a few, and those were the first two - questions from History that were blatantly on-topic there and had already been answered well and received a lot of attention.1 We'll take migrated questions, and if mods from other sites ask if a question would be good for HSM, myself of one of the other mods (Danu and LoganM) will give some input; we, the users of HSM are open to that. We have a decent % answered rate.
Here's where things get interesting. The arguments against migrating are that the intended target is a beta site and thus the questions may get less attention. Ironically, in many cases the result is exactly the opposite: Because we're a low-volume site, almost all questions are answered. On bigger sites, they could be passed over. Questions on HSM do have a better shot at getting answered.
So I'm indifferent on whether or not questions will benefit from being on a smaller site. But it's up to those on Mathematics to decide, not me. Anyway, this bit isn't the main point of your question. I think. By the way, though, mathematics is by far our most popular topic.
On to the specific questions.
To what extent were mathematicians in previous centuries aware of the lack of rigour in their methods?
I feel like this is a tad broad. The question is relevant for the past few millennia. There are many possible examples to explore, all with differing circumstances. If we got this, I might suggest that it be closed. A fine question, though.
When did the notion of the tangent space emerge?
We actually have a related question on HSM, though it has not yet been answered. There's a good chance that if the one on HSM is answered the answer will address the question on Mathematics, in which case it would be closed as a duplicate. But it hasn't been answered in three months, which leads me to believe that this (much simpler) one would be fine, and could in fact be answered.
Very nice answer by Famous Blue Raincoat. One of the main ways that a mod is very likely to migrate is if the OP explicitly asks for it. I just want to stress one of things written in that answer.
Please, do not cross-post. I can understand if the question is more than 60 days old or has already been migrated, in which case migration is impossible. But it's a pain in the neck to have to merge questions - because putting the same question on two or more SEs is not good. Here's the seemingly canonical answer to that (which actually partly goes against what I'm saying). Although Shog does say that under certain conditions cross-posting is acceptable. So what I'm recommending is not always the case. It's just my personal preference. Don't take it as SE policy.
Some sort of conclusion
I think that the Meta.SE post actually addresses the situation pretty well. Whatever is missing is mostly covered in Famous Blue Raincoat's answer. My two cents is that these posts would be fine on HSM (well, sort of) but that there's most likely not going to be a significant change in views; the math-history tag has a good % answered (~80%). I don't think that HSM being in beta will be a problem.
And once again, as Famous Blue Raincoat said:
I understand you want to help HSM, but when you ask a mod to close someone's on-topic question as off-topic, they are likely to decline.
Instead of flagging, leave a comment for the OP suggesting that they either repost or request migration to HSM, by flagging their question.
1In case anyone's curious, we've had 14 migrations so far. The first two were, as I said, rejected for obvious reasons. Two others were closed as duplicates and then merged because of cross-posting. Two were from Mathematics, and they've done well.