Frankfurt is home to a large internet exchange (once 1 of only 5 in Europe) where many different telco companies cables meet in a single building exchanging traffic with each other. Many companies systems is housed in the same building for short acces to that critical exchange. As I remember that building have 2 cable entrance points to ducts under 2 different roads.
An airlines systems might be redundant with standbys in different buildings but the question is always when do you switch to the alternative. In this case the main system was still up and running. Airline systems are also very interconnected so will you switch to the backup system if ops control in other parts of Lufthansa is running fine but checkin at a particular airport is not.
Checkin systems are also often airport supplied and standardised so the desks can be used by many different airlines. Then it wouldn't be LH's system that was down but the airports connection to LH's main system that is most likely in a different place altogheter. And if connection to and through the internet exchange would be the airports responsibility. It would be the airlines responsibility not to have a manual backup system. But these day airport flows are so system dependent that sample baggage handling couldn't be done manually without systems working and I do believe some planes took off from Frankfurt without baggage.