The PAX went online, provided a PNR, last name, email address, and checked into his/her flight. After these very deliberate actions, the PAX reasonably assumed check in was complete. It should not matter if they clicked a link in the email or not — they were checked in.
The PAX is not to blame for not clicking any link prior to arriving at the airport. If this was a requirement it should have been mentioned in the check-in process AND in the email. And, even then, the agent at the airport should have assisted a customer who had a valid ticket for the flight.
Although I can understand the benefits of having the PAX click a link, this is no different than a PAX losing their paper boarding pass — it happens and staff need to be prepared for it.