There are internal servers talking to other servers which don't need ipv4. There's zerotier that gives you a private IPv6 network regardless of your network capabilities. The only part that actually still requires ipv4 is the general public user.
Not always. Sometimes you talk between different providers, or to private endpoints on other services. Or its simpler use IPv6 between two networks rather than keep a global list of ipv4 ranges to make sure they don't collide.
Using IPv6 also comes along with many other complexities. If you already have a working IPv4 solution, working around costly public IPv4 addresses is likely much easier than trying to switch to IPv6 and having to adapt every part of your infrastructure to the many surprising ways it's different.
The problem of having to re-IP your entire network doesn't go away in IPv6, it just happens in different circumstances. If you're using public IPs as recommended by the design, then if you ever switch ISP you need to re-IP your entire subnet, just as you would with overlapping private subnets in IPv4-world. Of course, the same would happen in IPv4, but it is almost unheard of to use public IPs for all machines in a private network there.
There is very little extra simplicity in IPv6. The only thing it really does that most people fight with in any common place is to get rid of NAT, which is a fairly well understood technology by now. But getting rid of DHCP didn't work, peer-to-peer networks still get bogged down in firewalls even if they don't fight NAT, MTU problems are still around when running VPNs, etc. Maybe getting rid of ARP has helped in some scenarios for more complex networks?
Instead, you have to deal with both DHCPv6 and SLAAC, with multiple IPs on every interface, with much harder to remember IPs, with always changing IPs, with the myriad IPv6/IPv4 conversion schemes, with larger DNS responses (requiring DNS over TCP more often), and I'm sure I'm forgetting a few things.
... for all services.
There are internal servers talking to other servers which don't need ipv4. There's zerotier that gives you a private IPv6 network regardless of your network capabilities. The only part that actually still requires ipv4 is the general public user.