I had a small problem with installing the new CC, having to sign out and sign back in to CC to get LRCC to work. After that I thought all was fine and good, because I was able to install and use LRCC without a problem.
A few days later, I have noticed a couple of anomalies.
First, the Creative Cloud icon in the menu bar is almost always greyed out (it seems to show black at least some of the time, but only inconsistently). However, even when grey I can click on it and it does drop down correctly and shows my apps as up to date (I was able to install 6.0.1 yesterday).
Second, the history is far out of date. It shows the recent update of LRCC to 6.0.1 (22 hours ago) and then the next entry is listed as 372 days ago (Photoshop CC), which is WAY out of date! There have been many other updates since then.
It isn't clear that anything is failing to work correctly, however.