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The primary issue with Aquinas is that he's essentially pairing a "god of the gaps" fallacy with philosophical ideas that predate the scientific method we would need in order to functionally claim most of what he's talking about.
For example, he declares with confidence in his fourth way that because somethings are hotter, colder, etc. that there must also be an ultimate good just like there is ultimate heat. He begins the claim with scientific observation and then immediately rolls it into the field of philosophy and ethics. Now someone from the year 500AD might not consider that an issue since the scientific method didn't even exist at the time and all natural philosophy was on the same playing field, but modern people wouldn't consider those two fields to just be overlapping and logically interchangeable in that manner.
In the fifth way he claims that because certain beings have agency (or sapience, like us) and certain objects do not, that all non sapient objects must operate according to a being with said agency. This is patently untrue with modern scientific understanding as well, water flows because of friction and gravity, not because it was caused to do so by a god of some variety. Rocks fall, seasons change, etc. all due to natural processes. Not because there NEEDS to be a being with knowledge that guides it.
It's interesting because this claim is foundless as he hasn't proven that all objects operate based on a "plan" of some variety, he merely makes the claim that a plan from a sapient being is required for anything to happen and then begins to assess conclusions based on said claim. Moreso than that, it occurs in contradiction with his attempted understanding at potential and kinetic energy from the first way. He seems to have an idea about potential energy but then throws it out to just claim that objects or animals without knowledge operate on something else's will.
Thus beginning a long standing religious tradition of using scientific rhetoric where its helpful and attempting to shoehorn philosophy in where it contradicts or fails to uphold.
Ok so first off, thank you for typing out a well thought argument.
I posted a summed up version of the five ways, rather than the full text, and now I realize that probably was a mistake. I just wanted to make sure people would have read it, most would have ignored a wall of text. Instead, I will directly quote the full text in my answers here.
Here is a TL;DR, cause this will be long:
I don't think he tried to use scientific rethoric at all, nor that any philosophical shoehorning has happened. Rather, it's entirely philosophy. Doesn't mean it's perfect or necessarily correct, but we gotta call it the way it is. I also think you might be trying a bit too hard to interpret it as science, while that's not really what the Summa was meant to be. Some of your conclusions were drawn from the summary I posted not being accurate (sorry about that, btw) and I adressed them by quoting the full text.
Starting from the fourth way:
You correctly criticized his mistake in using fire as the source of maximum heat and mixing in scientifical evidence with philosophy, but the full text tells a more nuanced story.
Fire here is more of an example, rather than pure scientifical evidence. It's also not the basis of the point he is adressing here. That would instead be more abstract (and wouldn't you know it, philosophical) concepts like "good" and "true". So while your discussion on splitting natural sciences and philosophy makes a lot of sense, I don't think it applies here.
Onto the fifth way:
In truth, I think this is the most beautiful of the five ways and the one that, to me,makes the most sense from a scientific perspective. I remained of the opinion that Aquinas wasn't trying to bring in natural sciences into this one, but since you brought up "modern scientifical understanding" I will do my best to make some sense of it, according to modern science.
The message here is not as easy as water flowing because of gravity. It's also not as easy as "what was before the Big Bang?", because that would be, like you said, vulnerable to the "God of the gaps" counter argument.
Rather, starting from the universal constants such as the Boltzmann constant which regulates all of thermodinamycs; the speed of light in a vacuum, which regulates all existing radiation or the gravitational constant, which regulates how all matter and time interact; through science we get a very clear picture of how many pieces needed to fall into place for reality as we know it to come together, let alone life to be possible. According to this modern interpretation, the fifth way states that in order for the universe to exist as we know it, defined according to these specific constants, it must have happened through a higher being, a creator. Here, actually, is the only place where I see a possible mistake, because on a logical level he doesn't prove definitively that the existence of God is the only solution to the problem, the hypothesis of a coincidence remains on the table. However I personally think, when put in this perspective, the religious hypothesis remains the more believable one.
On your last point, I don't see how the fifth way would violate what he has established from the first way. The fifth claims that motion of inanimate objects happens naturally and repeatedly because of "some intelligent being [...] [whom] we call God". The first instead says that God was the first who put everything in motion, and that because of that things have been kept in motion ever since the universe began. I think these two point go hand in hand, rather than being opposed:
God first created the universe, by putting things in motion. God also defined the patters according to which things should have moved after his initial "push". This makes perfect sense to me.