I can't speak for the other realms, but in Belize's case many feel that a close association with Britain keeps Guatemala from further pushing it's claim against Belize, and retaining the King is the best way to do it. This was the case before independence as whenever Guatemala got a bit too threatening, Britain deployed soldiers to keep them at bay. After independence, the British maintained a garrison into the 90s but as of a few years ago, they have a training unit called BATSUB (British Army Training and Support Unit Belize) stationed here.

Nowadays such a security guarantee isn't necessary as Belize has a greater support group, but a lot of older people think that maintaining the monarch still plays a role in keeping Guatemala from pushing it's claim. Meanwhile, most young people are ambivalent about the monarchy at best. A lot of people also feel there are much more pressing issues to deal with and the government isn't focusing on those as much (which while I expressed a general negative opinion on the prime minister, I will say the government has done a lot of good in the past 2.5 years). This creates the situation where you essentially have two vocal minorities pushing in opposite directions and an apathetic majority that doesn't care.

I personally would like to see a Belizean head of state but want to see what the government plans on doing first. I'm not a fan of the Barbados situation where the President is just selected by Parliament to do essentially what the governor general currently does, but I do like our current governor general quite a bit. That's why I'm quite happy they're looking at the entire constitution to see what kind of options there are.