SilverCat38 I was able to solve this issue by installing an earlier version of pCloud app from Aurora Store. Had to try quite a few of them before finding one that allowed me to get to the login screen and log in. I think it was version 4.2 that finally worked (didn't downgrade version by version but jumped backwards a few at a time because I was getting frustrated). After I was logged in, I updated the app to the latest version and everything seems to be working fine. So apparently something changed in the app's code in the past few months that caused this issue of not being able to get to the login screen without Google Play Services installed.
I contacted the support team when this issue arose and they basically just told me to activate Google Play Services, which prompted me to ask them to pass a message to the dev team to consider making a version that does not rely on any Google APIs. Not holding my breath waiting for that to happen, though.