the speed test from PS4 isn't accurate either. It's a general quick shot test.
And also PS4 services have been slowed down for everyone in order to manage the load on the servers from everyone and their mother downloading large game files since everyone was at home for covid, and most still are.
But that's weird that you can't get a connection at all on your PS4 when you manually specify the DNS to Google.
Are you sure you did it correctly?
Last I checked you couldn't just do the DNS without also doing the IP manually as well (cuz the playstation be dumb like that and make you do ALL of it manually if you wanna do just the one thing)
So make sure you set up your static IP properly.
(Which is required for port forwarding also btw if you were to do that.)
If you don't know how to do static IP here's the gist:
Set the IP (might as well manually set it to what it was auto set to)
The subnet mask should be 255.255.255.0
The default gateway should be the IP of the router (the IP you use to get into the router settings on a browser)
Then you set the DNS manually.
For the Primary as I stated was 8.8.8.8 (make sure there's no spaces or extra digits. just the eights and the dots.)
For Secondary it's 8.8.4.4 (same thing, no extra digits, no extra dots, but don't forget any dots either)
If you do all this and you get no connection to the internet on your network, then something is very wrong.
Either your ISP is evil and is locking things down hella tight (possible, but not likely)
Your router settings are set up in a way that makes things impossible for you to do this kind of thing (should deal with that)
Or perhaps the country you live in doesn't like Google DNS? (maybe if you live in China that is)