The supply you propose will work well.

As to the cap across the secondary. If it is a very small value, .1uf or so it is there to shunt power line hash. Not a bad idea at all. A larger value is a "cheater cap" used to tune the output voltage.

A pure choke input filter will give you about 66% of your transformer secondary voltage. A cap input about 140%. As you increase the size of the cap you move from one mode to the other…John