Attempting to load a saved game currently leads to an error message complaining about the lack of the Myst disc in the drive. Clearly this is not supposed to happen and clearly, the game is more or less unplayable because of this.
Even getting the game to save without crashing is a problem most of the time. Disappointing, very disappointing.
EDIT: In fact, I can no longer even start a new game. The same "No disc in Drive" now appears, rendering the game completely unplayable.

I'm not having that problem...... what are your specs.... OS, 32/64 bit, etc.... quicktime/iTunes .... etc..... previous version of Myst installed?
Give us more details and we'll see what we can do.

XP Home (32bit). Quicktime and iTunes installed. No previous version of Myst installed. I will try reinstalling the game but not expecting much. I was expecting it to be temperamental with the Quicktime integration (which it is, tons of static, sound popping, etc). But having the game actually disc check seems like a problem on the part of the party that converted it into a downloadable title.
If you have any suggestions though, I would appreciate them. I pointed support to the problem, so if they send any news my way, I'll post it here for others to follow up on.

I see the quicktime integration issues with static/grainy quality-- which I easily forgive, just to be able to play it again on new machines......
I've been considering your save game issue.... I would almost suggest redownloading a fresh install copy to try your reinstall with-- maybe something glitched in the dl process. Make sure you let gog do their file integrity check also.
Other than that... what are the chances it could be a region issue? I see you're in the Czech Republic. I'm not sure how regions work exactly, but I know that sometimes game downloads go funky in Europe or Asia as compared to N. America. Any other games from gog give you problems before?

Nope, no real previous problems. The checksum check passed and all. Really not sure what could be the problem, redownloaded now and will see what happens.
Tried disabling compatibility as well, but the changes won't save. That is, I can seemingly apply them, back out of the dialogue, but when I check again, it's back to the default Win 95 compatibility mode.

that's odd.... but tallsasquatch is having a similar issue in the thread
http://www.gog.com/en/forum/myst_series/myst_installs_but_won_amp_039_t_even_load
"or at least the settings I saved did not stay checked after closing the window"
That seems to be an XP issue, there, I'm not too sure how to fix that. Have you tried unchecking the boxes, applying, hit ok, and then reopen the properties to see (without running the program)--- would be good to know if the changes are not being saved or if running Myst causes the default to be reloaded.

Have tried all the above already and no, the settings revert even without starting Myst. However, I noticed, that the only setting that actually does save is the choice of OS to imitate. I can not fully disable the compatibility mode, but can freely choose between 95/98/NT/2000. The rest reverts.
EDIT:
Tried reinstalling and saving and loading. This time I was able to resume successfully, though this seems fairly random. It's entirely possible for it to demand a disc again next time, and then every time after that. There seems to be some bug in the workaround for the disc check in the game.
However, saving a game or loading a game is still a lengthy process that takes upwards of a minute during which the audio stutters constantly. This also happens during most animated sequences, most obvious of which is moving the book case around, which causes the entire animation video to slow down and take far longer than usual while the sound constantly cuts in and out.

Gotcha... glad you made some progress. I'm not sure about the stutters and slow downs.... I can't imagine it being a RAM or Video Card problem (as the specs are like 256 MB RAM recommended and a 8MB video card..... But just to eliminate the possibility, make sure you update your video card drivers to the most recent ones.
I just installed Myst on my XP 32 bit machine, and I'm running into the same stuttering issues and cannot get rid of compatibility mode.... I'm wondering if the 2 are related. Will keep trying stuff and let you know if I get it fixed.