I have to admit I'm guilty of this, though it's not for the lack of trying.
I used to switch to dps when there were too many healers on the team, but nearly all of those ended really badly. Experience has shown me not to trust a 8k HP healer to keep a team alive, and there's always one or two in that situation. No vitality accessories/extra attributes clearly shows they are new to PvP and will most likely have a hard time handling the pressure.
It is most likely they queued PvP as a healer because they are most familiar with it when it comes to PvE. It shouldn't take them long to realize PvP is a different beast and either adjust to it, change class or simply stop. Them having other jobs at 60 doesn't mean they did more with them than reaching to the cap. It is very possible they are unfamiliar with them or have never bothered setting them up for PvP, and that's not something you can do in under a minute (have to fit all those additional skills somewhere and adjust bindings). I have everything at 60 and I haven't played over half of them since, much less have I took the time to set up more than a few for PvP.
In the end, they have their mind set on healing just as much as I do and that's why you end up with nobody switching. Does it suck? Sure it does. I can't say I've seen it happen nearly as often as you have mentioned however. I'm pretty sure the opposite situation where nobody wants to heal happens a lot more and I would argue that is worse.
There isn't much that can be done about it. The only thing that could would be to enforce roles, but I feel that would have a bigger negative impact than it would do good in the end.
- As Feast has shown us, some roles (namely DPS) would get longer queues, especially if you combine it with the current Grand Company restrictions that already make the wait pretty long for some people. You cannot compare this with how it works in PvE simply because the population isn't there to support it. The PvP community is way smaller.
- You lose a lot of flexibility when it comes to composition. Each map has it's particularities that make a global enforced composition unpractical. There are situation where a single (good) healer is enough, others where you'd rather have 2 or even more. You can absolutely do without a tank in Shatter to get a bit more damage on crystals. It would also prevent you from switching from a melee to a range in an attempt to balance your team. People rarely change job in the middle of a frontline to adapt to a situation (e.g. switch to a more defensive composition to hold an outpost at the end of Shatter), but it remains an option.
TLDR version: I don't think anything should be changed.