Which transformer faults will lead to power outage?
A transformer is inherently a fuse at the end of the day. All distribution transformers have damage curves. If many faults have occurred on a feeder and a transformer has seen excess current multiple times those damage curves change. Eventually the transformer will fail and it will fail open after some time if faults are allowed to persist.

Distribution protection schemes are geared at protecting transformers and other infrastructure. Generally, when a fault occurs (the vast majority are single phase to ground and caused by wildlife and vegetation) a breaker, recloser, or fuse up stream from the transformer should open before the transformer is damaged, or a fuse down stream from the transformer should burn up. All fuses have time current curves (TCC) and those curves should be arranged so they are beneath the transformer's damage curves. Fuses are cheaper to replace (~$25-50) than transformers and lines.

There is a lot more to it that I wont go into. But it all has to do with power company performance ratings and whether they would rather breakers trip to save fuses, or rather have fuses blow to minimize the number of customers affected by outages. These schemes may be alternated at different times of year, in different areas, and in different whether conditions.

So to answer your question in simple terms - most faults don't originate in the transformer. They can, but generally it is because the transformer has already been damaged. The fault generally originates on the line and may quickly end up in the transformer - e.g. the insulation is breaking down (faults can 'evolve'). Generally you're trying to save the transformer from faults to protect them from damage in the first place.

Of course, around here folks like to make sport of shooting transformers to watch them burn. That would be a good example of a fault originating in a transformer (seriously). I have no useful numbers on how many outages are caused because of that!

Here is a detailed article on the subject from cooper that shows  a fuse TCC curve and transformer damage curve

Edit: Looked back at your OP and saw where you said '300kV' which is generation/transmission class, not distribution. You still have zones of protection and will try to protect your transformer from faults on the secondary using downstream breakers and fuses. So some of what I said will still be true, but my answer isn't completely relevant. Unfortunately I know very little about transmission systems or transformer insulation breakdown so I can't edit this post to contribute much else. I'll keep this post here in case it contains some useful information or is useful to someone else.

I do agree with some of the comments before though about transformers showing symptoms of an impending failure. A transformer that is beginning to fail may not cause an outage at all if engineers can identify the issue and reroute power quickly enough before the protection equipment is engaged, a fuse blows, or the transformer burns. But it may mean a generator has to be taken off line or big substation air break switches need to be operated (which can be a big and costly effort and may or may not mean there will be momentary service interruptions).