HAproxy is good at what it does but it’s only good at proxying and simple rules. For the most part, it’s used as a load balancer and router and doesn’t really process the requests itself.

To add something here: HAProxy’s ACLs are more powerful than anything nginx, Apache or even Envoy can do. Of course HAProxy is not a web server but “just” a reverse proxy that speaks HTTP (and TCP) but what you can do with its ACLs is often extremely impressive in its simplicity and elegance. A single-line ACL in HAProxy would require loading additional modules in nginx and writing a screenful of configuration directives. Though the average self-hoster will probably never need any of the power HAProxy offers.

In the past 20 years I have professionally used all four of these as web servers and/or reverse proxies and I am pretty confident that HAProxy beats all others when it comes to request processing. Though Envoy might be getting there.

But what can it do more easily than an mid 20th century home can do?

One word: Automations. Everything from lights to irrigation, HVAC to surveillance cameras, fishtanks to plant monitoring, managed by a single, extensible open source platform, hosted locally in your own home.

Of course that is not trivial. If you don’t see your smart home as a hobby you enjoy putting time and effort into then the smart home scene is not for you, especially not Home Assistant.

Sounds like you are in a fine place with your home, so you are probably not the target audience here.