In the '60s and '70s, some bands added horns because they thought the music wanted them, some added horns after others did it and many of the horn tracks were less than great. Some used real Jazz musicians, some used good musicians who were able to get the job done while some were only used as accents. King Crimson's first LP had Ian McDonald on Sax & Flute- this was a British Prog band, before the name 'Prog' was used and it was a groundbreaking record. The sound isn't great on the original and even the remasters have sonic problems, but that's because the recording engineers had no idea how to record them since they had been used mainly for classical and other music that didn't have the same instrumentation, sounds and other details. While this music is definitely not for everyone because of the sounds that can be irritating, the atonality and because it can be described as 'on the edge', it has some great music that many other bands and musicians cite as major influences. McDonald went on to playing in Foreigner and I think you have heard him playing on their songs. While it's not his best playing, you can look into him online for more.
As far as the 'real Jazz' players, saxophone seemed to be the instrument of choice when a rock band wanted to scratch the edge of the "We're being creative" scab. Steely Dan always used outstanding musicians and if you know their Aja LP, you know that sax was featured on several songs, including the title track. That was Wayne Shorter, who's a great musician and has played with some of the best, including Miles Davis, Art Blakey's Jazz Messengers and others. If you're familiar with Billy Joel's song 'Just The Way You Are', that had Phil Woods on Sax and he's another one with a long and illustrious history.
There's a lot of Jazz that sounds like noise to many people but if you look into the history of this genre, you'll see that the musicians improvised because they were bored with playing the same parts, over and over. Rock musicians who don't just play three chords and fast notes without knowing if/why they fit the chords become bored, too- I suspect this is one of the reasons for their substance abuse, along with the famous ones being cooped up in hotel rooms and sequestered from the public when they weren't on stage. Some had healthy hobbies, so they had other things to keep their minds occupied.
Digging into any style of music can be a big rabbit hole and Jazz, being over 100 years old with many major shifts, is a good way to find out who did what, when, why and who they played with, as well as their other influences. Not that Jazz has the long history of Classical music, but its origin and where it has gone can be very interesting.
You mentioned Hard Bop as contemporary stuff, but that style actually came along almost 70 years ago.
Check out the early-mid-'70s Fusion stuff- this was much more energetic than a lot of what came along in the late-'70s, which was often called 'Fuzak' and it was a formulated cesspool created by the record companies' wanting hits. You can usually tell if the artists were in this era because they left the labels as soon as possible and many had legal fights over it.
One thing Jazz musicians have done is take parts of existing music (often Classical or the pop songs of the time) and wrote new melodies over the chord changes. One of of these is 'Ornithology', by Charlie Parker. He used the changes from 'How High The Moon'.
If you watch old videos of Jazz bands, they used sheet music- the early ones used it because memorizing everything would have been almost impossible but they also used it as the basis for their arrangers to leave space when it was time for a solo.
Here's an example of this, in a studio- he has been in bands like Art Blakey's Jazz Messengers and Phil Woods' Quartet, Horace Silver, Eddie Palmieri and even on recordings by Prince. Yeah, they're reading the intro and many other fast parts-
