That would be entirely up to SiteGround. It depends on how they've configured Innodb. CB is designed to work with MySQL 5.7+ defaults, which were provided above and below. It's possible SiteGround changed the defaults to reduce server resources and cut costs or are using older MySQL defaults (e.g. from 5.6 and earlier days), which could be a problem as you've experienced.

The below is all that should be necessary, but as I said earlier I am not a database expert and there could be other configuration that is causing problems.

innodb_large_prefix = 1
innodb_file_per_table = 1
innodb_file_format = Barracuda

ROW_FORMAT should also be set to Compressed or Dynamic.

If you are unable to use these settings you will hit row size limits very early and be very limited in the number of fields you can create.

You don't need a VPS for this. You just need a host that isn't stuck 7 years in the past still forcing MySQL 5.6 defaults, which will be end of life early next year. It looks like SiteGround announced an upgrade last year and in the comments it says they've MySQL 5.7 so I recommend contacting them to see if you for sure can get a MySQL 5.7 server or at least with MySQL 5.6 configured better.

www.siteground.com/blog/new-client-area-and-site-tools/

I'm not SiteGround so I've no clue. You'll need to ask them. According to HRISTO PANDJAROV in the comments of the article linked above they're rolling out new servers with 5.7. That was in August of last year. So I would assume something came of that at this point.

MySQL 5.7 isn't required. It's just the MySQL version that proper Innodb configuration defaults were set. With MySQL 5.6 the defaults for Innodb were very strict which limits data structure worse than MyISAM and if not changed then using Innodb becomes more of a burden than a benefit. If they do have the values in my above reply set then you should only need to change ROW_FORMAT for your tables to Dynamic or Compressed.

That's why I was asking if all I needed was MySQL 5.7, but it needs to be also set to:

Those are the defaults for Innodb on MySQL 5.7+ so they should be set regardless of the host if they note that they support MySQL 5.7.

Do you have any hosting companies that might allow this?

Any budget VPS will allow for this. I'm unsure why you said your VPS is too expensive. They're basically the same cost these days with cloud hosted VPS. For example Vultr offers $2.50/mo VPS packages that'd be plenty sufficient for a website. I'll break down the pricing for example below.

SiteGround StartUp - $0.33/mo for 3/mo then $11.95/mo with 50% off so $5.97/mo (if renewed for 12mo)
1 Website
10 GB

Vultr - $2.50/mo
1 Instance
10 GB

Both have your standard 1 click installs of various things. Vultr is a VPS so you get your own web server instance that you deploy. You can point as many website registers to it as you want. It's significantly cheaper than SiteGround StartUp once that 3/mo deal runs out. You could deploy multiple Vultr instances for that same price. You can use whatever MySQL version you want in your VPS. There's other great VPS hosts out there as well like Linode and DigitalOcean.

I don't really recommend hosting on anything other than VPS, but I prefer having completely control over my hosting environment. The shared hosting packages are attractive due to their prices, but those are typically only introductory deals like you see above. When those deals run out they're no cheaper than VPS and VPS lets me configure my server to do whatever I want. The major benefit of shared hosting of course is the simplicity of it which is best if you're handing the site off to a client.

Aside from that I don't know what to recommend. I don't host websites nor have clients as I'm entirely full time on Community Builder. I'm not sure why you're having issues on SiteGround either as we've several other users using them just fine. Maybe it's simply due to the server they put you on? I'm not sure. I suppose you could find hosts with prices that fit your needs and simply contact them to see if they either have the supplied configuration on MySQL 5.6 or offer MySQL 5.7.