I think LH lived up to its responsibility to get you to your destination even though you missed the separately booked flight due to the technical problem. Some carriers would not have gone that far since your contact for carriage with LH was only to get you to MUC in a timely and safe fashion. You don't mention if LH provided delay compensation for the initial flight. Did they provide meal vouchers and if necessary hotel rooms? Did they provide anything in MUC until they flew you out to your final [where was that?] destination.

I think covering some costs of the initial delay would be in order, but once you got to MUC anything else LH did was quite generous because they had no obligation to compensate you for another trip booked separately and not with LH. Maybe I am overly cautious about such things, but whenever I am travelling with such a connection to an unconnected next flight/trip, I always leave a day to cover such problems when the onward travel is non-cancellable.

So, no, LH -- or UA -- need not compensate you anything further.