Github is only free for public repos. You could use Bitbucket which does offer private repos for up to 5 users. Otherwise, your pricing is correct.
One note on Forge, it installs MySQL 5.6 and performance_schema is on by default starting w/ 5.6. You'll want to edit your /etc/mysql/my.cnf and add performance_schema=off. Performance schema will consume a lot of your available memory (20-25%) on that droplet.
You may encounter issues with a Forge deployed $5 Digital Ocean VPS as MySQL 5.6 needs 1GB ram and I think Composer might run into problems as well.
So, yeah opt for the $10 Digital Ocean VPS unless you don't need MySQL and/or are prepared to tinker with the server a bit.
You don't need Forge. Configuring VPS is not that hard (it's certainly much more easier than actually developing an application) and wasting $10 a month on it is unreasonable when it's more than half of your budget. Spend that money on Digital Ocean subscription.
In further regard to memory usage on a $5 droplet, Forge configures swap (I'm too lazy to check to see how much right now).
Of course Forge isn't necessary. However, it's so convenient and I'm soooo lazy. Plus, why not grease the man/machine that brings us Laravel? I kick in for Envoyer for the very same reason. I can do all that garbage myself, but unlike code, it brings me little joy. YMMV.
Forge is built if You Want to be Hassle Free Configuring the Server...
And Focus More On the App Development...
For me Forge is more like Using A Cpanel ...
You can Easily Add Site, Manage Cron...
It is Worth it :D