> I didn't know RDS had PgBouncer under the hood

I don't think it does. AWS has this feature under RDS Proxy, but it's an extra service and comes with extra cost (and a bit cumbersome to use in my opinion, it should have been designed as a checkbox, not an entire separate thing to maintain).

Although, it technically has "load balancer", in form of a DNS entry that resolves to a random reader replica, if I recall correctly.