Ignore the running of bgpd on the webserver - that's really an extremely specific use-case that is not useful for most folks.
However, imagine your scenario where you have two routers, two switches, and two load balancers configured in failover (LVS per your example) - with webservers behind that stack.
Now you need more than a single load balancer worth of capacity? How do you scale it?
Generally, you're pretty much stuck doing DNS RR to load balance across VIPs, and you add at least one VIP per load balancer you have. Need to do maintenance? Good luck not directing traffic to the load balancer you want to take out of service :) You can wait 3 days for all the DNS caches in the world to purge, or you're going to be killing sessions when you fail that VIP over.
Now consider instead running BGPd vs. DNS RR.
You have a single VIP, and as many load balancers as you like. I enjoy HAProxy, so I'll use it here.
haproxy01 1.1.1.1 haproxy02 1.1.1.2 haproxy03 1.1.1.3
All these machines advertise the VIP to the switch they are connected to. You set up path cost on your network so all these VIPs share an equal cost at your routers, and your router will ECMP to each. Doesn't matter what switch or rack your HAProxy box is connected to in your network (I suggest paying attention for traffic management reasons) - as long as it can speak BGP to the switch/router, the traffic flows.
Need to do maint? Kill bgpd on one of the HAProxy boxes. Current sessions stay up, you go get a coffee, and when you come back you have an empty session table and are free to do whatever you like to the machine. Turn it back on by starting up bgpd, and watch your traffic instantly rebalance.
You are completely correct that if you don't need more than one (in a pair for HA) load balancer worth of capacity, this solution is likely overkill. But once you need to start scaling, you're quickly going to learn that it will either be prohibitively expensive, or come with lots of downsides. There pretty much are no downsides to this architecture, other than needing someone with a small amount of clue to operate it.