No it doesn't. Not at all actually.
https://en.wikipedia.org/wiki/Twin_study#Criticism
Maybe an actual geneticist can jump in here, but it's my non-expert opinion that you all are jumping to conclusions about a topic that you're not experts in either.
The very first critique is basically a nitpick. Yes, figures will be overestimated, but not taking into account air resistance when dropping your bowling ball off the tower of Piza isn't going to invalidate a conclusion that gravity is an accelerational force of at least 9m/s^2.
In fact, with the exception of the statistical critique by the guy with a giant axe to grind it seems mostly nitpicking. Dr Axe Grinder seems to mostly be critiquing the additive model, claiming it gives wrong results, but more recent papers (such as Polderman et al (2015)) seem to say that the additive model is mostly correct. And given the latter paper is a many-author meta-analysis rather than an emotionally charged rant by a single professor and is much more recent, I'm inclined to go with the latter paper.
Your third paragraph makes me suspect you don't know how twin studies often work. The idea is that you measure the IQs of genetically identical twins and compare how similar they are to how similar non-identical twins are. This lets you isolate what is purely personally genetic.