I've been doing the exact same thing: using LR Classic for everything, basically, and just exporting those photos I want on my iOS devices to Photos on my Mac. Then (prior to Catalina) I would sync thru iTunes to iPads, etc. Generally it worked pretty good.
I really didn't want to get involved in Adobe's LR CC version and did not realize you could sync from the Classic version. Will have to check this out.
Like I said up above, once I updated to 10.15.3 I was able to sync through Finder to my iPad, with iOS 13.3. I have no idea what changed from 15.2 to 15.3, if anything, but it did allow me to sync. I had never synced using the "from folder" option in Photos, which is what I did. When I used to be able to sync from Photos/Aperture, the photos would appear on my iPad in the same order as they were on disk - I mean there would be a "Main" album for the year and sub-albums for each folder within the year. Now, they just all come in under the year - no subfolders. But beggars can't be choosers.
I also have a laptop and had Catalina on it until I put Mojave on it. There are times when both computers, running the same version of the OS would behave differently.
And I know this sounds crazy, but I have had photos just plainly disappear from my iPad.
I always like to think that each new version of the OS brings improvements and enhancements, and I have been very fortunate that this is the first update that has really caused me problems. I think Apple should "perfect" one OS before they attempt to "improve" it.
Interesting what you said about the "shared" albums. Does using them enable you to get your photos on your iPad?