The FRA check-in system will be one of the new SITA or ARINC Common User Check-In Systems. These do indeed work across a wide range of carriers, but they have to pull the data from the underlying, incompatible, systems.

If a QR check-in agent at FRA were to use the system, their login details will allow them to pull up details of QR reservations, others using the same underlying reservation system, and probably other airlines handled by the same ground handling company at FRA. While the system is CAPABLE of interfacing with multiple underlying systems, what an agent can actually see (and what they are able to do) will be restricted by their login details.

It's a little bit like being in a company and using a corporate IT system - the person on reception will login to the same system as the CEO, but won't be able to see the same level of information or have the same rights.

As for baggage problems, while the final carrier may well choose to help you out, it is the airline or handling company whose code appears on the baggage claim label who is responsible legally for your bag, not the final carrier. The baggage check label you receive at check-in bears the IATA carrier code or number for the company who is liable. So in the examples you give, the final carrier may help you out, but you can bet they will be claiming any costs back from the originating carrier.

In this case, the basic problems therefore remain:

1. The airline you are checking-in with has no easy way to verify that you have a ticketed reservation on the connecting flight unless these are on the same ticket/reservation. They are having to accept what you are saying (and any printout you are showing them) on trust.

2. The carrier whose code appears on your checked baggage label is ultimately going to have to pay the bill for any loss or damage to your baggage.