Assuming you don't need syncing across devices, the main drawback to self-hosting is that it only receives updates while your PC is switched on. Some feeds update often enough that you'll miss stories if you don't grab them multiple times per day.
This is a good idea. I use Thunderbird only for a small number of feeds I want to read every post from. I used to also use a separate feed reader for my "river of news", but eventually I stopped looking at that and just loadded hackernews or reddit when I wanted a distraction. But I might try Thunderbird for more feeds and just auto-mark most of them as read so I can browse at my leisure.
Or I don't know how