Well, you can’t get to a GPS using Newtonian physics alone because of gravitational time dilation. The clock in space runs at a different speed than the clock on earth.

If they tried, scientists could perhaps have come up with some sort of fudge factor (like what I think they do with dark matter) and made it work, sure. But it would have been really confusing to them when they figured out that all the clocks in space run at a different speed than the same clocks on earth.

Relativity plays a big role in astronomy too. One of the big questions in physics in Einstein’s day was why the planets didn’t appear exactly where you expected given Newtonian laws. Becoming spacefaring without general relativity would have been difficult.