the first question is, why can't these guys fix it???  you need to find
a different shop that knows what they're doing!!!

to answer the question, the car will still run, but if you're in the
habit of driving about with the light on, you're at significant risk of
damaging something expensive if another fault code develops because
you're already ignoring the fault indicator.  just get an aftermarket
catalyst and deal with the problem properly.

The short answer is "no" - your car will do just fine without the catalytic
converter working at peak efficiency. In fact, the emissions are probably
not much greater than they were before the light came on.

The long answer involves knowing what "codes" are being seen by the shop
when they read them. Although the codes are specific, they aren't always
clear and don't often point right to the specific problem... that's why the
shop has danced around things like this. In fact, replacing the catalytic
converter may not be the solution, and that's a good reason for them to
evade it for so long. If you can find out what the codes are (they will be
in the form of a "P" followed by 4 digits, like "P0452") and relate them
here, we can confuse you more <8^P

There are two downsides I can think of in having the light on, other than
the distraction. There is a small risk that another problem that would
normally turn on the light will now be missed, and I don't believe you can
legally sell a car in the US with the "check engine" light on (not sure
about that.)

Mike