1 The problem is somewhat more complicated in spirit to Goldbach and similar problems (FLT, ZFC): You certainly can invent a TM that solves instances of 3-SAT (in exp time) as you can check instances of <PERSON>.
2 You want to know however whether there is a faster TM and which is the fastest one. Since there are faster and faster machines (eg by recoding) none of which are subexponential though, this does not constitute proof of N=NP, neither does it yield a lower bound on time complexity.
3 Hence, what you can do is
(a) fix an encoding of 3-SAT formulae.
(b) test for satisfiability (in exp time)
(c) run "all" TMs (ie all algorithms, all proofs) against larger and larger instances of 3-SAT. these - (i) will provide wrong answers: discard - (ii) will provide correct answers: check time behaviour for larger and larger instances - (iii) will not halt sometimes, the (in)famous halting problem: keep in list and simulate further (will eventually drop out by exceeding any time bound).
From (ii) you obtain upper bounds on the timing behaviour. Discard any machines with $T(d) > n\cdot |d|^n$, $d$ the description of the 3-SAT instance, n a growing parameter.
Ingeniously mix TMs, 3-SAT instances, time bound parameter values $n$ ... 143
Eventually you have no machine left for a given fixed $n$, with $n\to\infty$ discarding P=NP, or else some TM (require $n > |TM|$ or so) is the solution for all say $|d| \leq n^{n^n}$, P$\doteq$NP. You are famous!
(Well, almost: you will have a machine that correctly decides 3-Sat in poly time for some (not so) small instances, $|d| \leq n^{n^n}$ ... and $|TM|$ far smaller - it is then up to you to mathematically prove the correctness (and speed) for any $|d|$. But you have an algorithm to begin with.)
Same problem here as with <PERSON>: Showing P=NP is OK (like a counterexample to <PERSON>), disproving it requires "infinite time" - you actually never disprove. You have an $P\neq NP$-checker (for a finite, but arbitrary large, set of instances!) as you have a <PERSON>-checker (which is what you are asking for in your question). 915