Can anyone advise me on the best dedicated/cloud hosting solution for 200+ WordPress websites?

The company that currently hosts these websites is WPEngine and while the service is great it is a bit pricey for what I'm doing and they have very limited functionality; such as no SSH access, limited WP-CLI access and a few other things which prevents me from doing some automated tasks that I would like to do.

My question is, would it be much more time consuming if I were to setup a managed cloud hosting solution to host these websites on a server a have full access to and what's the best way for me to prevent downtime? Also, what is the best environment for optimizing multiple WordPress websites so that everything loads quickly? I know WPEngine uses both Nginx & Apache and PHP with HHVM and here is the phpinfo:

http://yonghi.wpengine.com/phpinfo.php

Here are some statistics for all of the websites:

• Total Storage: 20612.9 MB

• Total Visitors (Dec 21 — Jan 20): 254,880

• Total Visitors (Jan 21 — Feb 20): 257,091

• Total Bandwidth (10 Day Avg): 975.330933 MB

• Paying: $600/mo