I used to use Pocket. Like OP I had an extensive collection and experimented most of the described issues. For me the solution was to migrate to DEVONthink. It

I used to use Pocket. Like OP I had an extensive collection and experimented most of the described issues. For me the solution was to migrate to DEVONthink. It supports WebArchives for saving whole pages but also a few other formats that can preserve only content. Its search is great. It also has an iOS app. It’s web clip browser extension can save the page as you see it (most of the time) so can save pages behind a paywall or login. I don’t think there are apps for other platforms so probably not a solution for the OP.