HAproxy is good at what it does but it’s only good at proxying and simple rules.

It’s possible to write very complex rules/ACLs with HAproxy… stick-tables, ACLs with regexes on whatever HTTP header, source or destination ACLs, map files, geoblocking, lua scripting, load-balancing from round-robin to host header load balancing, dynamic backend servers provisionning through DNS… Not that you can’t do it with Nginx (it started as a reverse-proxy before becoming a jack of all trades), nor that nginx isn’t a great tool (it is!), but HAProxy can do very complex things too. It also follows the good ol’ UNIX philosophy of “one program to do one thing and do it well” and thus doesn’t try to be a webserver, hence why you need a webserver behind it to serve anything from static files to PHP/Python/whatever.

If you are in France, or around Europe and don’t mind sending your Pi via mail to them, Faimaison and Tetaneutral do propose small computers hosting in their datacenter racks, Pi type included, but also NUCs, respectively for 24€/month (bit expensive but small structure compared to Tetaneutral edit: it’s ~15€/month nowadays) and 5 to 10€ / month. That’s just an example. Generally you’ll get one IPv4 and one /56 or /64 IPv6 prefix.

You might want to look near your location if there is a LUG, non-profit ISP, or non-profit colocation proposing the same kind of services. You may even meet some nice people! But it’s definitely doable at least in Europe.