'shared hosting' ... any provider ... is not normally good hosting for a moodle ... due to the need for resources of a moodle site. You only get slices of the pie. And as you have already discovered, can't do things one might need to do.
But then you say "managed cloud hosting service" ... managed means the provider does the heavy lifting ... if it's a dedicated server and NOT shared then they should do what you request within technical limits of what you have leased - space/memory/processing power/etc.
All services on one server ... web server + DB server (localhost) is a memory balancing act between web server handling connections and making request and the DB server - it's ability to respond.
Haven't seen phpmyadmin tools but do know MySQLTuner shows important information about the localhost DB server and databases that are significant for config and DB performance.
Clips from a live K12 site 7493 users - 160 courses - using Tuner:
-------- Performance Metrics
[--] Up for: 4d 7h 25m 9s (17M q [46.249 qps], 94K conn, TX: 20G, RX: 2G)
[--] Reads / Writes: 92% / 8%
[--] Binary logging is enabled
* [--] Physical Memory : 31.1G
* [--] Max MySQL memory : 13.8G
[OK] Maximum reached memory usage: 4.6G (14.85% of installed RAM)
[OK] Maximum possible memory usage: 13.8G (44.25% of installed RAM)
[OK] Overall possible memory usage with other process is compatible with memory available
[OK] Slow queries: 0% (20/17M)
-------- InnoDB Metrics ----------------------------------------------------------------------------
[--] InnoDB is enabled.
[--] InnoDB Thread Concurrency: 0
[OK] InnoDB File per table is activated
* [OK] InnoDB buffer pool / data size: 4.0G/2.9G
[OK] Ratio InnoDB log file size / InnoDB Buffer pool size: 512.0M * 2/4.0G should be equal to 25%
[OK] InnoDB buffer pool instances: 4
* above is important
Does phpmyadmin give you such infomation?
'all websites' ... on same server?
How many? And what are they? Do you have a WordPress/Moodle mix?
Then there is consideration of clients to your moodle. The above clips came from a K12 entity that sometimes uses the Moodle synchronously ... teacher F-2-F with X number of students who are also using a course in the moodle at the same time.
Your clients? ... asynchronously? ... pop in at any time ... there are no set 'class times' - where instructors/teachers are to be online same time as particiants in a class for whatever purpose ... exception would be zoom (other class meetings).
So how are courses conducted in your Moodle?
'SoS', Ken