I dont believe that. Bench pressing and bw dips got me to doing pushups like a machine. Before the heavy metal in hs I struggled with push ups.
Im almost certain that is false.
The workouts aren't mutually exclusive.
You are doing both, lifting weights and calisthenics. Thats pretty much the goal.
A lot of people think you can only do one. Which is not true. You should do both.
There's plenty of cats who think because they can bench 300 that they don't need to be able to do 25 pushups. And one plenty of people who think because they can do 100 pushups in one set that they don't need to be able to bench at least 225. There's nothing wrong with having a preference toward one or the other. But you can very much do both, and it sounds like you do.