Hello,
One thing you can try is crawling your site with screaming frog and get a list of all the urls, with said exported 404 you can check in most link checkers (e.g. OSE, Majestic, Ahrefs) via bulk checkers,with this you can roughly work out the highest value of those links as a way to prioritize your work, not the best solution but it may at least give you some where to start.
Regards to 404's Google does check them think of it like this, its looking through a site it will follow a link and if that links results in a 404 it would be marked down but next time the crawler comes to the site it will do the same thing lets say you've fixed that 404 it would then crawl that same page and find the new 301'ed content of however you fix it etc.
finally don't look at the pages as 404 look at them as lost content and add a value to that, if you have a missing page that in fact is a gem of content that was once useful and people are still clocking on that link looking for the same content then its defiantly worth trying to fix that, however pages with little content or value may not be as helpful it is always nice to have a "clean" site with 0 404s If you decide to not fix the 404's its not the worst thing in the world it's just a possible leak in link juice but if you're not fussed don't sweat it, especially if you can put the time from fixing thousands of 404's into creating awesome content etc.
Hope some of that helps, and good luck and happy 20th Birthday to your site!