I don't see the problem.
I'm sure there is a mode where the PS4 can download updates when it's turned off.
Another solution would be to play offline. No updates needed.
And I find it hilarious how people still bitch on install times. You aren't forced to sit in front of the console and watch the install bar to go up. Play or do something else! It doesn't take a huge amount of time.
Also doesn't the PS4 now also support external HDDs? Just plug in one of those and keep all of your favourite games there. I have got around 20 HDDs flying around. Many aren't even in use as they are too small for my needs and I've updgraded them to 4TB and 5TB.
Still, even small ones that are only 500GB or 1TB are enough for that task and can be found for example in old laptops.