The posts and articles with good backlinks, does that content still make sense in your renewed site? If so, I'd bring them back. If you don't have the content, you can try the Wayback Machine. The same goes for any old posts you think would be useful to your new readers.
The problem with redirecting a bunch of 404s to the same page (like the homepage) is that you end up with soft 404s and not a very good user experience. Pick the ones that correspond to specific pages that you have on the updated site and redirect those to the equivalent page.
Anything else, I'd let 404. A bunch of old posts, with no good links, the content of which you no longer have a use for on the site don't represent value to searchers—those pages will just drop out of Googles index (and crawl attempts) over time.
[This isn't just theoretical. We changed domains back in November and we had lots of old content—going back 10+ years, which is ancient history for a financial publisher. I ended up with about 6,000 404s. We are now down to about 4,000 404s as pages drop off. Google crawls us quickly and regularly and our organic traffic is up 86.49% .]