Where should I repair my iPad's broken screen?

Repairs are best performed by Apple or an Apple Authorised Service Provider (AASP) - as only they have access to genuine parts.

As you correctly observe, repairs by independent repairers may cause some elements of your iPad to cease working - such as TouchID, FaceID or the iPad's cameras. If repaired by anyone other than Apple or AASP, you will forfeit all future technical and repair support by Apple.

You can establish eligibility and cost for an iPad repair by Apple in your region/country here:

iPad Repair & Service - Apple Support

For practical purposes, in lieu of repairing and returning your own iPad, Apple usually exchange the iPad for a renewed/refurbished iPad of the same model and specification. The cost of the exchange is the same as would otherwise be charged for repair. Unless you have benefit of an active AppleCare plan, an out-of-warranty repair while expensive will be somewhat less than the cost of a replacement iPad.

However, given the age of your iPad, the relative cost of repair/exchange may be uneconomic. You might be better advised to consider putting this money towards the cost of a current model iPad - therefore benefiting from much newer and more powerful hardware with a longer potential support lifetime.

You're very welcome.