How to remove a jammed, broken key from a lock

They've done the exact modification that you're proposing (placing it next to the door, rather than in the door) at my place of work.

What you're looking for is called an 'electronic strike'.

You cut out the area around the existing strike plate, and wire in the electronic strike. You'll need to feed it power, and a signal from whatever you have to grant access. (in our case, the card reader, plus a motion detector so it'll open from the inside automatically; you don't need the motion detector if you still have a functioning door handle from the inside)

A quick search online suggests that they run $100-500 (US$), but you'll probably want to contact the manufacturer of the card reader to ensure you get something that's compatible.

...

As for 'best practice', much of it comes down to cost vs. what you're trying to protect.

The advantage of the electronic strike is that it can be hard-wired for power (no batteries to replace every few months), and you don't have to worry about cable bundles through the door or wireless signals (for auth & logging) that can be intercepted or interfered with.

If you're only protecting a single door (so don't have a centralized authentication server), don't need to log each person individually, and don't mind changing out batteries once in while, you might try looking at the locks they use for hotel room doors that replace out the existing handle + lock mechanism.

Since you have a straight shot at the latch because you removed the molding, use a reciprocating saw and just cut the latch. Then you can replace the lock set.