Cloud saves are not out of the question for Reforged.
SCR, which was released 3 years earlier than WC3R, had a primitive cloud save feature added to it. From what I’ve seen so far, it saves settings, hotkeys, collection loadouts, and saved games (although there is some wonkiness with the saved games because it doesn’t take profiles into account when deciding to upload to the cloud or not).
I think the absence of cloud functionality in Reforged is more a result of the unfinished, rushed out release. We know the devs had more plans for Reforged a month before the project was scrapped (i.e. the news blog showcasing profiles). It’s possible cloud saves were on the docket, so it possibly could still be implemented… once the mess is cleaned up.
As long as the file structure hasn’t changed since release, and that the achievements are saved alongside the campaign progress, a workaround can be found here: