Interesting! Ctrl+R is indeed reset in TIC-80 but it shouldn't delete any save progress (e.g. when I try it right now the progress stays). Maybe some weird bug in in the engine or some setting your browser had coincidentally clear it?
I hacked together a save editor real quick. It's available here: https://gate.itch.io/ghost-save-editor?password=ghost-save-editor
To use it, just launch that page and check the puzzles you want completed in your save file. Exit that page, and when you reload the main game, the progress should be there.