I just followed your example and found the same thing. Weird thing is: This collection doesn't save, but the original HD versions I bought way back when DO save progress. I know it's down to the Flash cookies on that end, but makes me wonder how this collection works.
Viewing post in Submachine Complete Collection comments
I emailed Mateusz himself and he gave me these instructions:
" right click on the game [as in on the game window after launching], go to settings and find local storage there. enable some, 1 Mb is more than enough. exit, restart the game, should work."
(brackets are additional context)
I tested it myself, it works.