Hello @rocketman and welcome.
Kudos for doing your due diligence and learning about the MySQL search deprecation. I recently pencilled together some instructions on how to install Elasticsearch for a video. However, the Oracle Java License had a huge update and now “technically” you can’t use their JDK without paying for it - Which is a requirement of Elasticsearch, so the video never happened. Anyway, I digress…
5000-6000 monthly visits probably doesn’t warrant going with a Managed Dedicated Server, so I won’t talk about that any further. So, the thing is with Managed Shared Servers is that they’re incredibly cheap and easy for Web Hosts to run and maintain because they have a “cookie cutter” model. They all have these “glass ceilings” in place where their customers can’t modify a whole bunch because it makes it super-easy for their engineers to maintain.
If you’re running a Wordpress site or something similar it really doesn’t matter. But when you work with more complex CMS applications like Magento, you quickly find yourself running into hurdles. Therefore, Managed Shared Servers are generally very cheap.
Then you have your Managed/Unmanaged Cloud Servers (kind of the middle ground between Managed Shared and Managed Dedicated Servers). These are typically what I try to push most people towards as the setups allow for much more flexibility but don’t come with as much “A1 support” (at budget price levels) because each server is bespoke and comes with their own unique challenges.
Having said that (and I can already hear myself sounding like a cheap shill), I recently did a review on Cloudways. Essentially, you’ll probably find a similar experience as you’ve had with Siteground but with a little more flexibility of what you need at only a slightly higher price. My review video will give you a better idea of what I experienced: Cloudways Review: 30 days undercover as a customer (Magento 2 Managed Hosting).
I don’t think you’ll find anything else comparable in both price and service to Siteground that gives you the flexibility and support you’re after without spending much more than you do now.
Hopefully, this response helps shed some sort of light. I’ll be glad to go into more detail if you have any specific questions.
Anyway, its late - Have a good weekend.