Does the battery replacement help with performance issues?

What is the current Maximum Capacity of your battery?

Settings > Battery > Battery Health & Charging > Maximum Capacity

If the Maximum Capacity is 80% or less, then the battery needs to be replaced.

A new battery will bring another 2-3 years of use with your phone with improved performance, so if your current phone has the features that you need, it would be a good investment to buy a new battery.

Yes, it should. As to whether it's worth it, it is significantly less expensive than buying a new phone, and the XR IS compatible with iOS 17, due out shortly. In all likelihood, the XR may not get iOS 18 or whatever follows on next, so if you want to stay current with iOS releases, replacing the battery now will probably get you through another year.

Assuming a new phone is announced soon, the price of the iphone 14 will probably come down a bit, assuming it's not discontinued. You may want to hold out to see what shakes out before making the decision.

Will the XR last another couple of years with a new battery?

See my previous answer above. You don't have to worry about the XR "lasting". The battery is the issue.