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It was used by reservation; that is, someone requested a time slot and got access to the computer for that time (paying the required fee, of course). IIRC, it even ran overnight.
I saw the system while I was a pre-teen, in 1982. At the time, it was still in operation even after more capable systems were installed and microcomputers were making their inroads. At that time its usage was reduced, but CNC folks told me it still was used by some civil engineers who did structural calculations with it.
(I've built two online systems for teaching my students computing: https://bcp.cs.montana.edu and https://mtmc.cs.montana.edu w/a similar vibe)
Add ten more years, and the IBM 801 could have been a CPU architecture good enough to scale all the way to the present day without emulation, unlike the 360.
Even after somewhat "mass market" systems, the software was almost always entirely custom for the end-user.
Apparently the current attempts to throw LLMs at the problem are running into the issue that there's very little open source IBM 360 code available to train on.