What “TfL trains” are you using that aren’t part of the oyster system and accept QR e-tickets?
Also how is buying a paper ticket faster and easier than just tapping your card at the gate line? You can even wait till you know your train is going to arrive before tapping!
Both Oyster and National Rail tickets are valid on the Elizabeth Line, despite it being effectively part of the TfL network. Should National Rail introduce a smartcard? Probably - but they don't have one yet, so it's not unreasonable that parent would be frustrated by the unreliable QR code scanners.
16 zones limit would not have existed in the first place if they did not insist on keeping all the fare info on the card. Many fare systems do fine with online-only operation.
Also how is buying a paper ticket faster and easier than just tapping your card at the gate line? You can even wait till you know your train is going to arrive before tapping!