Couple of things:
1. How long have you lived at the place?
2. Is the system serviced by an HVAC company at least once a year (once for heat, once for cooling)?
3. Who has the previous service records?
Frankly, the HVAC company’s explanation that a missing filter has caused $1000 worth of damage is suspect. Unless your air is filthy, it would take more than 5-6 months without a filter to totally screw the coil. Also, are they saying the coil is clogged, or are the claiming its corroded? If its clogged, shop around for an HVAC company that will wash the coil for cheaper… $400-$500 is a reasonable price for cleaning a coil. If its corroded or leaking, then I would absolutely have a reputable service company come out and do a 2nd opinion, because that sounds like a crap explanation.