You would have to be pretty unlucky to get a midwife that pushes the button and then stands back waiting for the docs - not saying that it cant happen though. As its part of our mandatory annual update, all midwives should be capable of dealing with SD, and in fact, in all but the most severe SD (in almost all the hospitals that I have worked in), it is generally the midwife that resolves it, with the team being used for assisting to get the woman into the mcroberts position, and with suprapubic pressure etc.
However, there are very few genuine shoulder dystocias, most are easily dealt with by the manueveres as above, and they can mostly be dealt with at home as well as in hospital. What cant happen at home is the level of resus for baby if it is a rare severe SD, as we only carry fairly rudimentary resus equipment, and if it takes 5 mins or so of manipulation to get baby out, then is going to need resus.
There shouldn't generally be any more blood and tearing etc than normal - in many years working as a senior labour ward midwife, and dealing with literally dozens of SDs, I have never had to perform an epis for access and it is purely a mechanical condition where the top shoulder is stuck behind the pubic bone, so once it is turned or manipulated into a better position, it shouldn't cause any more damage than would have been seen otherwise.
It can happen with any size of baby, but it is true that statistically, bigger babies are slightly more prone to causing SD, but the worst SD I have experienced was only 3.4kg and I had delivered her previous baby which was bigger with no problem.
Sometimes tbh, some doctors (and midwives) will use any excuse to get you into hospital.