Yay, something I know something about! Firstly, yes leaks in the coils is very common. As highland said the metal is very thin in the coil to increase the efficiency of the system. Second, it sounds like he never actually identified the leak. I know from personal experience it isn't always easy to find a leak but until you know where exactly it is located you cannot estimate how much it will take to fix. If the leak is in the tube bends of the coil, most likely if it is the inside coil, then it can be a pretty quick patch and done. But if the leak is in the middle of the coil then it can be a gamble as you need to cut away the fins in the area to expose the tubing with the possibility that the leak cannot be repaired. Where I work we do commercial and industrial HVAC service and we will not even attempt to repair a leak inside the fins, it just isn't worth trying. Residential service techs are much more likely to attempt the repair, and may even have a decent success rate, but their first job is to sell you new equipment.
$3,000 to repair a leak in a coil is insane. Assuming it is a repairable leak we would charge someone around $1,300, and that is only if they needed a hard quote. That would include 8 hrs labor, refrigerant and a filter. If someone is telling you it would be $3,000 then the next thing they will say is that they need to replace the unit. But I'm guessing that you have an R22 system which isn't sold anymore, thanks obama, and they would have to install a whole new system, condenser and fan coil.
My suggestion is to find a local one man company. Forget any large service companies as their job is to sell equipment not to fix your problem. The guy in a truck as we call them can be happy making $800 in a day to do a simple repair instead of trying to sell you on a whole new system.
It is entirely possible you have a freak oddball leak. Meaning it is not in a usual place and/or very hard to detect. I had a cracked nut on my unit last year that took several hours and multiple technicians to find as it was only detectable when the unit was full of refrigerant and running. The time spend finding the leak can add up. You may be able to do some self investigation and try to take a look at your coil for oil residue. The oil will come out with the refrigerant but it doesn't evaporate and will be a dead giveaway that you have a leak.
Others have said to have it filled and see how fast it leaks out. You can do that, and I personally would but I can get the refrigerant at cost ~$15/lb for R22, but if it is a fast leak than a) you wasted money and b) if it leaks fast enough then by law they have to disable your unit. "Fast enough" basically means if it would be empty in the next two weeks you must disable the unit. Residential prices for refrigerant are stupid expensive I have heard it going as high as $100/lb which given the cost above makes a nice bit of profit. Your best bet is to pressurize the system with nitrogen and listen for the leak. Nitrogen is cheap, very cheap.
I honestly cannot tell you why the residential HVAC market gouges customers so badly.
Project Manager/Estimator
HVAC & Plumbing Service
TL;DR
Find a different service guy preferably self employed.
Either visually locate leak or do a pressure test with nitrogen.
Hoplefully find a repairable leak and get it fixed for less than $3,000.
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