I cannot find Game Center on iPhone X

yes that's the correct section. If it's not listed after Podcasts, I'm not sure what to tell you. You could just start trying things like restarting your phone (directions on how to do a force restart in the link below)

https://support.apple.com/en-us/HT201412

I did see some old threads from people using iOS 10, but I think that was when Apple moved if from being an app to more integrated into the iOS in settings. One of those suggestions was to sign out of your Apple ID and then sign back in. you could also try the path of backing up your iPhone, resetting to factory settings and then restoring. A bit of a process though.

How to back up your iPhone, iPad, and iPod touch - Apple Support

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

Or you could try contacting support and see if someone at Apple has a solution for you

Get Support

It should be there... You have to scroll down a bit.. can you find 'Music', then below that TV, Photos, Camera, Books, Podcasts... and then Game Center? Literally the last Setting before it starts to list your individual apps

Are you running 13.6.1?