Bounds on the Twin Prime counting function Maybe I'm mistaken, but wouldn't it suffice to show that the twin prime counting function has any kind of strictly increasing lower bound to show that there are infinitely many twin primes?

I think that you have the answers you want.
But the real answer is no :
$\pi_2{(x)}>\frac{2x-1}{x}$ for every $x>5$
Clearly, $\frac{2x-1}{x}$ is strictly increasing but this tells us nothing for twin primes.
