I understand your situation. I believe if you don't want to wear an old outfit get an extra advance and rapid server from siteground or godaddy. Both are having a lot of happy customers and the prices are also good comparing to the market.
Yes, you've read correctly. PHP 7 is much faster than previous versions.
A week ago, I was also searching the same.
There are lots of hosting that provide PHP7 hosting.
I've found many hosting providers provide benchmark of php 7 on their hosting.
Few of them can be found below. You can read their blog.
I'm with Cloudways since Nov 2015, found it very optimized, 1 click application installations and they've integrated advance cached technologies like NGNIX, Varnish and Memcached to boost page load performance.