If you look at the countries that consistently do best in the tables drawn up by Amnesty, RSF and so on, they tend to be:
Monarchies:
Norway
Sweden
Netherlands
Denmark
Republics:
Finland
Iceland
Possibly a hands-off monarchy is the thing to have; Canada, Australia and New Zealand tend to do a bit better than the UK, despite having the same nominal monarchy; but I don't think they were running government bill past Prince Charles, as mentioned above. The UK normally comes out in the same area as similarly sized European republics - Germany, France, Italy. Or it may be that smaller population sizes help, and that's why those listed above do better than the 'big 4' European countries.
As for the OBE, knighthood etc. system - you can have that without a monarchy (the US has Congressional Medals of Honor etc., France the Legion d'Honeur). Very few of them are actually decided on by the royals (there are one or two special classes for people in the Royal Household). Britain has more grades of awards than most, I think; I think it is a good idea, overall - a community does feel some pride when a head teacher gets an OBE for 20 years' outstanding work, or whatever it is.
The most contentious part of the awards system is peerages, of course, because that (currently) puts them in the House of Lords, able to legislate. In theory, we will soon have a nearly-all-elected House of Lords, but the laws to implement it just seem to get delayed. They've got rid of almost all the hereditary ones now, which is good; they still have the seats for Church of England bishops, which is bad (and I think the current proposals for electing Lords are going to leave them in, incredibly). In the life peers, ie those awarded a peerage in the honours system, you can get a few specialists that it's good to have in there to be authoritative in discussions (eg Robert Winston, pioneering fertility physician). But too many are retired politicians or businessmen (who typically were donors to the Tories or Labour). I want it to become all-elected, with maybe a few spaces for co-opted specialists.