Just my 2c I mean 2p.
Not all AM (MF) broadcast antennas are insulated at the base. you can use a grounded base and shunt feed her at the 50 ohm point.
I believe the vast majority of non-directional stations use a 1/4 wave or 90 electrical degree stick or close to that figure.
Transmitters do indeed have a built-in output network but you still require an antenna matching unit at the base of the mast or tower.
A carefully designed AMU is critical to optimize b/w & efficiency. It is not unheard of for two identical RF chain stations to sound completely different to one another without correct tuning. One can sound as if they have a pillow over the mic and the other, well you get the idea.
In a past life I was part owner of a small town AM in Williamsport, Pa. 1kW day and 38W night. We had a 229ft ND stick. Protecting a station in NY was the reason for low night power. I junked the old LPB 'low level mod' night TX in favor of a pulse width modulated solid state transmitter and nearly doubled the signal coverage!
We sold the station to a local girl and I moved back to the UK.
David G4ZOW