Something which I also take a little bit of issue with, is the title of this thread... "IT Support For Your Company (No Pay - No Way)". Whilst I realise that in the context, there is every chance that your IT company has not done it's job properly, there is certainly many a person who feels that IT companies should not charge for every minor thing.
To that I say, well a good IT person can easily command 6 figure salaries. Such people are expensive to hire (speaking from experience, both in hiring others, and if you wanted to hire me, I don't come cheap, that said I have no shortage of work), and you really need to ensure that everything they spend their time on is billable. So this does mean that the quick fix needs to be charged for, since half the reason why it is a quick and easy fix is your drawing on many years of industry experience. Furthermore the good ones are often able to anticipate issues before anyone else in the business knows there is an issue, and fix it. Which pretty much means that the really expensive IT person whom seems to charge for everything might actually be the reason why you haven't just spent the entire week with no computer systems.
As the OP may be discovering, there is a reason why the good ones are so expensive, and that is because the cheap ones are not worth the paper they print their resume on. I've also have the displeasure of hiring (and firing) and working with the not so good IT people. They are the ones whom don't solve the minor issues before they have become major problems, they are the ones that do spend a day working on what should have been a 10 minute fix. They are also the ones which might seem cheap on the surface, but if you then look at the fact that you've just had an outage lasting a week with your employees spending that entire time looking blankly at the walls, well it ain't so cheap any more (best one I know of was a outage which cost the company around the $250,000 mark, all because they wanted to save a couple of dollars)
That's not to say that you don't every so often come across a cheap IT person who is also good, but they are actually a pretty rare breed.