Glad to see this thread on the forum, as I've encountered what are, for me, GAME BREAKING problems with both the backwards compatible versions of Halo: Reach and Halo 3: ODST.
The issue is the same with both when playing on my Xbox One S. Neither game will save campaign progress while played offline.
I've installed both games from disc - they're pre-owned Xbox360 versions that I bought way, way back.
In each case, because my internet access is often limited, I've had to download the games in chunks. I'm not sure if that would make any odds to the way the games behave after install. In the case of Halo; Reach, I flagged the problem with Microsoft Xbox support. Their staff were very pleasant to deal with but completely unable to help, unfortunately. We managed to rule out user error, though, which is probably significant. Frm what I can gather, it seems this might be one of those sneaky tech problems that affects some Xbox One owners but not others. It's bizarre!
Reach WILL save campaign progress for me if I go online with it. But if I attempt to resume play offline... well, no option to resume the campaign appears whatsoever. So it's right back to the start every time. Offline, the game will only allow me to play with a temporary profile.
I've only just looked at ODST, and given up in disgust. That game recognises my Xbox profile, but still won't allow me to restore campaign progress after quitting the game. In the case of both games, I've been using the 'save and quit option' to leave the game. They're just not having it.
This is EXTREMELY frustrating. The campaigns are where it's at for me. PLEASE fix this if you can, 343. These games and their stories are too big and too good to be hamstrung with a requirement to be always online just to save progress - whether intentionally imposed or not.