For a hosting company just to turn off access to your site with no warning is very bad in my opinion. Installing wp-rocket should give your site a pretty huge speed bump and your visitors go up even higher with your pages loading quicker.
If anyone is looking for a good hosting company I have used siteground for years and they are excellent. I am not affiliated with them at all but they are fast, with excellent support.
Moving your website can be a bit daunting but there is an excellent free plugin called "All-In-One WP Migration" that makes it all very easy.