Technical issues with Heroes III.

the game simply cannot be installed there, in the sense of installing regular programs; instead, an archive with the game is downloaded, ready to run.

Here lies the first reason for the problems: ideally, the game should be installed so that the corresponding entries are made in the registry, which are necessary for its normal operation. If you simply copy the game to another PC or extract it from the archive, you can bypass this by finding, downloading, and running a REG file that already contains the necessary lines. I happen to have one, although I installed the game myself. I'm attaching it to the message – I hope it helps.
The second possible reason for the problems: you write that an HD mod is already included in the game in the archive. This is convenient for lazy (or previously unfamiliar with the game) players, but it is unknown what the "craftsmen" who "created" your archive might have changed/deleted.:) It is better to search for and find the game installer – even if it is a repack, they can be quite good (as luck would have it)).