You might be wrong to base your assumption of the garage on a replacement tyre cost. They are likely to be quoting for a premium brand (you needed to ask). The £46 tyre you had fitted is not.

Worn rear brake pads/shoes does affect the performance of the handbrake, so don't rule these out.

You need a full brake check at the likes of kwik fit who will strip the brakes down and inspect and give you a written report.