Consider that if Mozilla hadn't lied about how it was going to open source it when they announced they were buying Pocket (alternatively, if people actually held them accountable instead of demonstrating that the public's short memory means that you can lie about pretty much anything), then someone would have probably fixed at least a few of those.

If all you're doing is using it to store stuff, then instead of signalling to a company that bad behavior pays off by rewarding them for it, consider using Zotero instead, which doesn't have a history of being insultingly mendacious and by default keeps your data private (in the form of reliable, bog standard files on your own disk). Or, if you're willing to suffer through slightly more geekery, there's Perkeep/Camlistore.

I'd ideally like to move to something else but I haven't found an alternative that's as good. Not being able to export my data, do any kind of advanced searching, or access a few of my files in Pocket really does suck.

I see Zotero has a third-party Android app, maybe I'll try that and see if it is decent.