It's not problem of web servers in general, but specifically Apache.

HTTP keep-alive connections are faster (saves 3-way handshake on subsequent requests and allows pipelining).

The problem is that Apache implements them in a horribly inefficient manner (thread/process and all its memory kept in use just to hold on to a socket).

If you use nginx, lighttpd or shield Apache with haproxy/varnish, then you can easily have keep-alive enabled and clients will see better performance.

There is nothing wrong with keepalives if you give them proper timeout and request limit values. They definitely cut down tcp/ip setup overhead, and improve page render times. Just don't let the client be too greedy about how long they hold that connection.