I don't know, New Zealand's pretty keen on it's monarchy, all told, but trust in the government? yeeeeeeeeeeah, no. about the only parts trusted to be honest and compitent are the judges and Maybe the tax-men. and even then only when the politicians aren't actively fiddling.
if anything, one could make a case that the monarchy is liked in part because we Distrust the government. dare not let the 'elected' officials or beaurocrats have the additional power allowing them to control the Head of State position would grant. seriously, for everyone who actually Wants a republic (not just 'eh, a republic would be better, maybe, in an ideal world, but i don't see how you'd make it work' but actively would work towards doing it and seriously trying to get other people to support such an idea) you'll probably find someone who thinks a hereditory aristocracy of one sort or another replacing the current setup is a good idea. (well, that or a seperatist of some sort, that's a whole other kettle of fish.)
'course, most of the 'true republicans' are corporate types who long for a US level of plutocratic control of the government which they can't actually pull off under the current system (if only because NZ's too small to be able to Hide such a thing taking place.) and while there might be arguments and disputes between business owners and workers/unions, Noone trusts forign corporations, except aparantly a few of the elite at the top of the bigger NZ based corps and people like John Key (oh, the nicknames He's earned, both before and during his term....) ... Interestingly, even the Prime Minister is usualy addressed by his personal name, and spoken of by name.... oddly, the Queen isn't. (wouldn't surprise me if many people didn't even know her name, though everyone knows who she IS.)
also, should be noted: republic just means 'without a hereditary monarch' ... so.. I'm pretty sure you could find a Lot of countries which don't have actual monarchies where the government is trusted, and vice versa.
(well... i suppose we trust our government to be reletively Consistant, at least... but that's a different sense of the word.)
hummm.... i think i lost my point part way through that. oh well. gained it again by the end, i hope.