Can using a new evap coil on top of an older furnace cause problems?
The only technical reason that this isn't possible is that the coil needs a specific amount of air to flow through it when it's cooling. So to select the correct coil, you need to know the CFM of the air moving through the air handler. Not impossible to figure out or measure, but might not be easy for an air handler as old as yours.

However, if I was a professional HVAC installer, I would not want to put a new coil on a furnace that old, either. The installer only touches a small portion of your system, but in the eyes of the owner easily becomes responsible for the whole thing. So if I wasn't comfortable with my assessment of the existing air handler, I would probably decline to replace just the coil, too.

tl;dr: find a different installer if you don't want to replace the whole thing.