Yes, I agree with you that you have some connection issues at your end to work through. Until you get those issues ironed out it's probably better not to make assumptions as to the faults in the/a game's code. The following addresses your comments in the OP...

With GOG games being DRM-free, there is no reason for online/offline modes such as we find in Steam, since a good portion of Steam's games require Steamworks to be up and running before those games will run--that's Steam's most common form of DRM. (If the game uses Steam DRM then it will not run at all unless Steamworks is first up and running and the user is verified to have legitimately purchased the game.) OTOH, I don't own a single GOG game that has to have Gog Galaxy running before the game will run from its own executable--that's because none of Gog's games use Galaxy as DRM like Steam uses Steamworks as DRM for Steam's DRM games.

A very odd mistake I see from time to time is unfortunately repeated by people who buy GOG multiplayer games who mistake the game's online requirement for DRM, when in fact the online requirement is only for multiplayer play via an Internet server--and it has nothing whatever to do with a GOG DRM scheme. This seems a common and very puzzling mistake, imo. With GOG games--I own 358 of them according to GOG--I never, ever set online/offline mode--for any reason, since there exists no reason to do so.

I download and update all of my games through Galaxy, but run the games from their own executables with Galaxy disabled and turned off--not running, etc. All my game's run great. I really have no idea what is so confusing about this very simple fact: No game that you buy on GOG requires you to run the game through Galaxy--not a single one. All of them will run from their own exe regardless of how you downloaded and installed and/or updated the game.

It's really difficult to understand how and why people constantly confuse Galaxy with Steamworks--with Steam, Steamworks is often involved with DRM as a condition of running your game whereas the use of Galaxy with GOG for installing or updating is 100% optional--because again, GOG has no DRM requirement whatsoever. GOG also allows game owners who wish to do it to store and keep their own DRM-free installable copy of their game through manual download. Steam offers no remotely similar feature at all--Steam controls the games, those with DRM and without, the owner of the game copies (that's us) do not. Unless we are wise enough to prefer Gog's DRM-free service.

Your GOG games do not require online/offline modes to be set, whether you use Galaxy or you do not. If a game is multiplayer only, or includes a multiplayer mode, then *of course* you must have an Internet connection in order to play the game--which has absolutely nothing to do with DRM.

Some people seem incredibly confused on this very simple, easy-to-understand difference between GOG & Steam. I think the main reason people confuse the two services is because they understand Steam DRM pretty well, and they mistakenly think that Galaxy and Steamworks are exactly alike and perform the same functions. They definitely do not, I'm glad to say!

I do not understand reason to discuss DRM/Galaxy when original post is about online/offline mode implemented directly into the game (not any game, but specifically Hellpoint).

When Hellpoint starts and there is an internet connection present (just connection, no need for Galaxy) it goes into online mode.
And if you create character while having that “enjoy your stay” message, it means that save will be online ONLY. You cannot play offline with that character.
Starting the game without internet or when EXE is blocked by firewall will turn it into offline mode.
ANY ONLINE save will be unavailable; option to load game will be disabled.

This information not presented in game in any way. There is no warning about locking players to online mode after they started the game with active internet connection. Saves not marked as “online only” in list, instead game works like save do not exist.
And it leads to misunderstanding. People start their game (even without Galaxy, if they do not plan to play online), play for some time. Then start the game without connection (for example technical issues) and cannot load their save. First assumption – save files lost. If game has multiplayer mode – normally it requires internet connection for playing that mode. If the game requires constant online connection for single-player offline functions – reasons behind that are not exactly “obvious” (well… they are). Yep…