The only thing I would have done differently for my first change would be to capture the drained fluid and measure it so that I could replace the same amount of fluid as the original factory fill.
It's very simple to do yourself.

True dat about the fluid. But be careful modifying with external coolers or filters. I once installed an external cooler on a 3rd gen that ended up ruining the AC condenser. Sometimes it's better to trust the engineers knew what they were doing.

I had my transmission serviced recently at my local dealership and did a flush without replacing the screen. That’s the recommended service for the transmission. It was about the price you were quoted too. I thought that the transmission had a filter and was ready to replace it as normally done with filters, but it’s not required. There is a screen that is there which would have been much more invasive and not necessary. They’ll do a drain and fill if you want, but I wanted all the fluid replaced. I’ve had other transmission flushes done with other vehicles that were “non serviceable” without any issues too. Don’t worry about the details. I also regularly do most service myself, but some things I don’t like doing.
There are plenty of things we eat that we have no idea about how they were made.