> I wonder how profitable it actually is.
Apple doesn't need for these services to be very profitable. So long as they break even, investing in these platforms invests people in Apple's eco-system and hardware. While Music has an Android client and TV+ has some support outside Apple TV, many of the services require Apple hardware.
> Apple TV and Arcade have, by my understanding, not done great.
It's really hard to tell how good Apple TV+ is doing. Creating a streaming service/ network out of thin air is ridiculously difficult even for a big company like Apple. While I wouldn't pay $10/ month for Apple TV+, having it as part of a bigger bundle makes it a solid add-on.
I haven't tried Arcade because I don't game much, but having it as an add-on to a bundle means I have a source for casual games which are add-free.
I kind of suspect both Apple Arcade and Apply TV+ were launched last year to gain a little momentum for launching the bundles this year. The fact that TV+ was free for a lot of people over the past year until now supports that idea. As I said above, I wouldn't have paid for TV+, but I do enjoy some of the programming on it. (Ted Lasso is pretty damned good)