See, before the cloud, you used to deploy your code to actual physical servers. If your site became really popular you needed more than one server to run it.
Before commercial and open source load balancers became a thing we literally used HTTP redirects to distribute load. You might visit www.example.com and get redirected to www13.example.com, which was an actual box in a datacenter.
I don't think the internet would exist today if websites just randomly fell over once the number of simultaneous users exceeded the capacity of a single server.
We'd have certainly adapted if we couldn't share cookies, but it would have been a lot messier.
So no, you cannot reliably control load balancing using one domain name pointing to multiple IP addresses. (IP anycast is better suited, but much harder to implement)