So to confirm, this settings problem is effecting more than just Google logins?
Have you tried rebooting your Wi-Fi router, or using a different Wi-Fi network, or using cellular if that’s available with this iPad? Try a different Wi-Fi network if you can. That’ll eliminate the network as a source of problems.
If the problem is happening across networks, and if this problem is effecting more than just Google, and given you’ve rebooted, then I’d next try reloading..
Connect to a Mac or Windows computer, back up your iPad, reload iPadOS 13, and re-load from your backup. This backup—if it completes—can also be a potential path for recovery should this iPad have a hardware problem.
iCloud can also be used for backups if that’s been enabled, but that tends to be much slower than a local backup to a Mac or Windows system. And if it’s not been enabled, it doesn’t yet exist, which’ll make it even slower as it’ll all have to upload. (And if the network isn’t working well, or if there’s a hardware problem with this iPad, then, well... it might not work at all.)
How to back up your iPhone, iPad, and iPod touch - Apple Support
Restore your iPhone, iPad, or iPod to factory settings - Apple Support
The following are supplemental articles, and probably won’t be needed here.
Restore your iPhone, iPad, or iPod touch from a backup - Apple Support
Restore or set up your device from an iCloud backup - Apple Support
If you can't back up your iOS or iPadOS device to a computer or restore it from a backup - Apple Support