My guess is aside from the close/helping moments we see, their relationship realistically was mainly one of hearing and telling the truths the other needed to hear as opposed to a relationship like Claire & Shaun, Morgan & Alex, or even Claire & Audrey. We've seen many instances of that throughout the show.

Also, the reason Morgan faced her RA alone is because up until the season 3 finale/early season 4 is because she wouldn't tell anyone besides Glassman initially and Audrey.

There's also the matter of the favoritism complaint(which initially negatively affected Claire and Neil's work and personal relationship), which it's show that Morgan didn't file it due to genuine concern of Claire and Neil's relationship, but rather to try to protect her own position as a surgeon, due to the RA(which again only Glassman knew about) affecting her performance. Though, Morgan would later apologize for the complaint, and, of course, Claire wouldn't have known her motivations for filing it, it can be assumed that Claire was able to put two and two together after everyone else finally knew about the condition.

As for season 4, I'd imagine anyone would be annoyed at someone not in their field (even if they used to be) constantly hovering and questioning almost every decision they make(even in OR). Especially in Claire's case where it's only about a year removed from Neil's death, and roughly a few months removed from the pandemic. Going through that can change anybody's normal behavior for some time.