A three-winding transformer can be seen as two transformers in one and presenting a short-circuit level that will be dependent on the coupling reactance between the two secondaries. For ex., two generators feeding one common bus and with only one transformer has the least transformation cost, but the highest short-circuit level on that bus. The same generator, each one with their transformers present now the lowest Icc, but the highest transformation cost. With the 3-winding transformer, the cost of transformation should be in-between the two previous solutions, and the short circuit level will be dependent on the design of the transformer.
For applications where the two secondaries are used in traction power and rectifier applications, they present a high coupling (and the same voltage) which increases the short-circuit in the secondaries ("tightly-coupled stucked secondary" design). For applications to serve continuously equal loads in both secondaries, the loosely-coupled stacked secondary design is more appropriate. But if the loads are unbalanced for long periods, this design is not appropriate. With less coupling between secondaries, the less short-circuit will be available in the secondaries. There are other aspects of design impacting in more or less fault duties and losses, which is very well presented and summarized by a SKM's publication.
I've seen the unloaded delta tertiary in "Y-Y-delta" transformer designs, with the primary Y with solidly grounded neutral. The primary Y neutral point at HV side must, in general, be solidly grounded to provide a SLG short-circuit level to be able to trip the overcurrent protection of the power company. The secondary Y neutral point at MV side is grounded by a resistor (ex. 400 A) to limit the damage caused by SLG fault at the consumers side. The Y-Y windings alone would make any unbalance or the existence of 3rd harmonics from consumer side to pass through to the HV side unless there is this delta winding which traps this unbalance of loading in the form of a circulating current in the delta. So, this delta tertiary works as a zero sequence filter. There are studies held by power companies which compare the advantages and disadvantages of this kind of transformer design with the Y-delta design. There are cases where the tertiary has auxiliary service loads of the substation.