It depends on whether or not the freewheel and chainrings are significantly worn. If they are, then what he told you is true. It's hard to spot if you don't know what you're looking for.

I'd start by checking the chain myself. Measure it with a ruler. I don't know what the digital chain checker is, but most chain checkers greatly overestimate the wear. If you measure with a ruler and it's as bad as the shop's tool claimed then the chances are good that your cogs and chainrings are worn too.

I don't know all the possible signs of wear on cogs and chainrings, so I'm not sure I can be exhaustive. On vintage components, the teeth were generally symmetric so it's easier to judge. Once they started ramping the teeth it got more complicated.

The freewheel is a cheap replacement. The chainrings probably aren't, particularly not the big one. Upgrading the crankset and bottom bracket together might make more sense than replacing the chainrings. Is yours Octalink? (If the crank is a 5500-series 105 then it is Octalink.)

If your bike is aluminum and the rear spacing is 126 you might have trouble upgrading to a modern groupset. Also, if you have a freeewheel and not a cassette you'll need at least a new rear wheel.