Wow, if a am your tire shop i would be happy. Tires if in good condition can last more than 10 years, my Yokohama geolander already over 7 years still in good condition, if no serious crack and treads still within limit should no issue just let a real tire professional to check.
See Michelin website, it did not mention 5 years must throw away only 10 years max.

https://www.michelinman.com/auto/auto-tips-...-need-new-tires

1. Keep five years in mind

After five years or more in use, your tires should be thoroughly inspected at least once per year by a professional.

2. Ten years is a maximum

If the tires haven't been replaced 10 years after their date of manufacture, as a precaution, Michelin recommends replacing them with new tires. Even if they appear to be in usable condition and have not worn down to the tread wear indicator. This applies to spare tires as well.

If dunno, go and get proper advise and not being scam.
A lot of times those tire shop will sell your used tires and earn double the income if still in good condition.