I have just this very second, run some tests on my iPad Air v1 using:
(a) the in-built factory speaker
(b) a pair of in-ear headphones (not apple brand) plugged into the devices headphone socket
(c) a kitsound boobar Bluetooth speaker (Rechargeable)
I ran the same piece of music (Peter Gabriel track) in spotify and then in Apple Music. I can assure you that there is hardly any noticeable difference between the two apps, I am only using the free version of spotify so I don't have access to 'extreme' stream quality so i ran it on both normal and high settings.
I can honestly assure you it's very difficult to notice the difference between the two apps. They are practically the same volume and quality... I have even just got my wife involved with the tests and she can't notice the difference in volume or quality at all.
Perhaps wait and see what others have to say here on this forum, but I can't see any issue to complain of. Do you have more than one device you can test in case it is a volume setting or hardware issue?
I'm assuming you know about the setting in restrictions etc. aswell as volume limits and EQ settings and it's not just a single slide-bar to control the volume.
Anyhow let's see what others have to say, but for me the Apple Music quality and volume is almost indistinguishable with the output from spotify. I have not tried the other apps/services you mentioned, at this moment in time.