Please confirm if you also see no way to save / load the game in the remastered version. Are there any ways to go around that?

Nope. You have one save and that's it, just like the mobile version. No real way to go around that IIRC, unless you can copypaste the save files elsewhere (not so sure about that one).

Not familiar with the version, but if it doesn't write the save back into the exe file itself (believe it or not, few really ancient wonders did indeed exactly that), but produce some file(s) that can be located and copied that's always an option.

(I for one, have a large collection of FTL saves. Yes, that game also only ever write continue.sav on exit, and destroy that file on load, so I manually rename and move files around. Yes, I'm a habitual cheater.)