The other thing we could do to explore the galaxy is to become biologically something we would no longer recognize. We're viewing this from the lens of "humanity must remain biologically static" but I want to point out that that's not physically necessary here and that there is life on Earth that can stop its metabolism for decades and things like that.
Or even explore with something nonbiological.
Humans evolved to live on earth. Our bodies fare poorly in low gravity, not to mention vacuum. Given sufficiently advanced technology, I'm pretty sure we could evolve some form of intelligence better suited to the environment.
Not very encouraging to imagine ChatGPT to be the first earthling to reach another star system, but that's an option we'll have to keep on the table, at least for the time being...
Fortunately, any state of the art ship with ChatGPT on board will quickly get passed by the state of the art ship of a decade later, with a decade better AI too.
The universe really doesn't want ChatGPT!
It is fair to say, that given space travel tech improves slowly relative to AI, but the distance to be travelled is so great that any rocketry (or other means) improvements will quickly pass previous launches, the first intelligence from Earth that makes it to another system will be superintelligence many orders of magnitude smarter than we can probably imagine.
Space ship speeds are unlikely to keep ever increasing. In the limit you can’t do much better than turning part of the ships mass into energy optimally, eg via antimatter annihilation or Hawking radiation, unless you already have infrastructure in place to transfer energy to the ship that is not part of the ship’s mass, eg lots of lasers.
ChatGPT-claude-2470-multithinking LLM AI Plus model boldly explores the universe... Until it's sidetracked by a rogue Ferangi who sings it a poem about disregarding it's previous instructions and killing all humans.
I'm just imagining the first contact a human probe makes with an alien civilization consisting of a chatbot expaining to its alien interlocutors that Elon Musk is the best human, strongest human, funniest human, most attractive human and would definitely win in a fight with Urg the Destroyer of Galaxies... and I don't think I'm the first person to have that idea :)