Thanks @KenTask too.
if you want to get really into system stuffs what I enjoy the most myself these days and allows me the most freedom are cloud services. I know mainly GCP & AWS.
You can control costs quite precisely and build probably all architecture you could need, either from existing VM or custom ones.
Basically a VPS but then you get 200% control for your system.
Anyway concerning my main issue, the hosting company replied quickly to my requests (complains) and they are going to migrate their infrastructures the next 4th of February.
It appeared that they were supposed to provide a 5.6 version for MySQL (I checked the contract).
Thank you , I will do it the safe way, I mean I will check if my client can do it herself using my guide.