Probably shorted turns on the winding that got overloaded by the shorted MLCC. I've got several vintage HP TE with transformers that have failed in similar ways, one had the primary winding burnt open (from a trapped wire).
Some AC voltage measurements of the secondary windings might reveal if the output is lower than expected. No load voltages should be higher anyway, if another forum member with this DMM could help with normal reading that would be very helpful.
I'm assuming it's too new to have a detailed service manual?
Having said that, you definitely need a replacement transformer.
David