I don't think it would be possible to run the site off a "modern" shared
hosting plan. We are using one of the larger Linode VPS instances. We might
have to upgrade or similar here.
Some of the companies that I would have considered good at one time because
they offered things like SSH, FTP access, and similar in later years
discontinued those offerings or restricted them to 'by request' only. VPS is
just awesome. While the early companies offering VPS were pretty good
generally speaking there are a lot of companies now that I wouldn't use. I'm
not a fan of dishonest companies and terms like 'unlimited' are really great
way of turning me against your company.
I have had some experience dealing with shared hosting from GoDaddy (beyond
by my control) and found that there were so many restrictions it wasn't
possible to even run Drupal (at least not the way the person who was trying
to run it needed it). I had advised against them using GoDaddy. Unfortunately
they didn't listen. I basically looked into the problem they were having and
discovered what they wanted to do wouldn't work with GoDaddy's shared hosting
solutions.