I was able to locate the following discussion (and a few others that were marked duplicate of this one) on node discovery in Bitcoin:
I guess, the primary challenge is, the number of nodes is not under any one's control. So, a traditional load balancer wouldn't know in advance the number (and IP address) of the nodes. Thus, it wouldn't know how to balance the load. (Or, maybe it can figure out dynamically, not sure).
But, suppose, I wanted to launch a private version of Bitcoin where I had control over number of nodes that can ever be online. Then, I could set-up one URL on AWS ELB, for all clients to connect, where this URL will have all the nodes 'behind' it. Note that, it doesn't necessarily have to be AWS ELB; it could be the HAProxy or others.
Is this feasible?
For a detailed list of types of nodes, see this wonderful graphic from the book, 'Mastering Bitcoin'.