Is it possible to find a Mersenne Twister seed given only the first output?
Is it impossible to go from the first output to the seed, or is there some trick that I’m missing?


Yes, it is impossible, but not because the algorithm could not be reversed. If you had the whole state, you could go back to find the seed, but you need at least 624 32-bit outputs before you can uniquely find the state of MT19937 (due to the $k$-distribution property).

As @yyyyyyy mentions in the comments, seeding with a timestamp cannot produce all the possible states. However, due to collisions there will not always be a single 32-bit state that produces a particular 32-bit value as the first output, so even if you could work through the math to invert to the seed, you might not find a unique timestamp that way.

Your spoilered approach is much better.