It would absolutely be a privacy concern if someone without the rights to access this data could access it from the computer.
My understanding is that it’s the same account logged on both devices. Computers are multi-users devices. No technology ever would protect your secret stuff from someone you’ve just shared your personal account with.
It’s a problem that deletion is not perfectly synchronized, yes. It certainly is a privacy risk because an unauthorized intruder could find them. But in this particular case, there’s no intrusion. The wife just had normal access to these messages in the first place.