I personally like the fact that there is someone, who has reigned for 60+ years and seen parties come and go is there to protect us. While she doesn't exercise her powers. She is there as a safety net, and offers impartial advice (as she isnt party bound) to the current PM.
She is the last line of defence if a government decided to go all crazy on us.
That is largely how I see it.
Having someone who is not elected, doesn't have to worry about looking good for the next election, and is largely above party politics (or at least never speaks out about it, unlike many of the politicians who would prostitute themselves to get a sound bite on TV*), is quite a good anchor/stabilising influence on the elected politicians.
I doubt the Queen would ever use her veto unless there was no other choice, and if she didn't something seriously bad would happen, however I suspect that at times she's able to say to a politician "is that really a good idea, or are you just being a blithering idiot after a few votes", and suggest they rethink an idea).
Having a democracy without an unlected head of state is all well and good, but it leads to situations like in the US where basically no one can get into office, and stay in office whilst making decisions that aren't popular, but may be needed for the good of the country.
It's one of the reasons I'm marginally more in favour of the House of Lords as it is now, than as another elected house - you just have to look at the States to see how badly that can go wrong, and how it can paralyse the governments ability to pass legislation.
*Seriously, given how some of the politicians who are meant to be qualified lawyers are willing to spout rubbish that even I as someone with no legal training knows not to be the case (IE when they talk about changing the law to allow for you to defend yourself in your home, or when they try to score points because someone from another party quotes what is legally correct but unpopular), when it will make them look good, does not show them in a good light imo,