Will Flash games still work after 2020?
Generally speaking: yes, Flash games will still work come 2021 and beyond, as Adobe's decision to sunset the platform simply marks the end to official updates and patches being developed. The whole point of the move is for Adobe to cut costs by calling off the longest undertaking in the history of software development expeditions to the land of diminishing returns. It's hard to imagine a reason why Adobe would finally patch up such a sizable hole in its budget, only to create another one by funding some hypothetical task force that would scour the Internet for Flash content and demand its hosts delete it at once.
Then again, this is the same company that thought Flash will beat HTML5 in something other than the likelihood of being the attack vector used for stealing your social security number within the next 12 months, so we'll downgrade that scenario from a "never" to "wouldn't put it past them but fingers crossed" rating.
Flash games will still work post-2020 as every other alternative costs money
In other words, you'll still be able to install existing Flash plugins and extensions, then use those to access countless games developed for the said ecosystem over the last 25 years even beyond 2020. Every other alternative costs money, and Adobe is seemingly sick of hemorrhaging currency on Flash.
With that said, many existing repositories of Flash games will be shutting down in the coming months, so there's a chance you'll have to resort to hunting down mirrors of your favorite titles in the future. In case of any obscure favorites you may have, the safest bet would be to download them manually yourself before that happens - there's no shortage of free online tools allowing you to do so.
Alternatively, most existing web endeavors dedicated to archiving Flash games and other apps allow you to request individual titles and aren't too picky about what they approve. Yet even with all those efforts, some Flash games will inevitably end up lost to time, much like the sanity of whomever Adobe was paying to maintain this dumpster fire of a project in recent years. For all its faults, Flash was still the norm for web content across nearly three decades, and its dev community was churning out countless games on a daily basis throughout the entirety of that period.