The tyre trade can sell tyres up to two years old as new. Check the DOT code for week & year of manufacture, i.e. a 4 digit number e.g. '2605' means your tyre was made in week 26, year 2005.
The only check or recommendation I could find relating to "old" tyres is:
"Any tyre showing clear signs of wear or ageing ("cracking" or "grazing") must be examined by a tyre professional, even if it's un-run or little used, for example, spare tyre, tyres fitted to trailers, motor homes or caravans."
I think your tyre fitter is just trying to make a few quid. Most tyres will last a lot longer than 3 years and will not crack or graze if they're properly looked after.