What makes nginx/apache a web server, HAProxy not?
What functionalities HAProxy lacks to be a web server?
HAProxy can listen on port 80 and can speak HTTP but that's not what people mean when they say "web server."
HAProxy is not a web server, because "web server" implies an HTTP endpoint that can serve static content from files and/or dynamic content generated from code. That's not what HAProxy is for.
Technically, there are certain capabilities in HAProxy that can be misused to emulate some capabilities of a web server -- you can serve very small static files from memory buffers and you can generate small dynamic responses using the optional embedded Lua interpreter -- but it is not intended or designed to be used as a web server. It's a proxy server -- emulating a web server toward the client, and emulating a client toward the real back-end web server(s) behind it -- because bidirectional emulation is commonly what proxies do.
With Nginx and Apache, you can specify a root directory from which files are served, and you can specify paths that are to be serviced by code running in languages like Perl, PHP, Python, etc. Not with HAProxy, because, again, that isn't what it's designed to do.
Both Nginx and Apache can also be used as proxy servers, as HAProxy can, but HAproxy is specifically designed and optimized for that primary purpose -- proxying and load balancing against multiple back-end, selecting the back-end using various rules and algorithms... in essence, HAProxy is an "intermediate router" for HTTP requests, delivering them rather than responding to them. It can also proxy and load balance non-HTTP protocols that rely on TCP.