Why not just use DNS load balancing over VIPs served by HA pairs of load balancers?
Back in the day we did this with Netscalers doing L7 load balancing in clusters, and then Cisco Distributed Directors doing DNS load balancing across those clusters.
It can take days/weeks to bleed off connections from a VIP that is in the DNS load balancing, but since you've got an H/A pair of load balancers on every VIP you can fail over and fail back across each pair to do routine maintenance.
That worked acceptably for a company with a $10B stock valuation at the time.
Back in the day we did this with Netscalers doing L7 load balancing in clusters, and then Cisco Distributed Directors doing DNS load balancing across those clusters.
It can take days/weeks to bleed off connections from a VIP that is in the DNS load balancing, but since you've got an H/A pair of load balancers on every VIP you can fail over and fail back across each pair to do routine maintenance.
That worked acceptably for a company with a $10B stock valuation at the time.