Should a store have a search bar?

Above about fifty products, yes and prominently. Search users convert at several times the rate of browsers because they arrived knowing what they want.

Why the answer is what it is

Which means search quality matters more than search presence. A search that returns nothing for a misspelled brand name is losing the highest-intent visitors the store has, silently.

What this looks like in practice

A product page has one job. Everything on it that is not helping somebody decide is in the way.

Consistency beats cleverness. A customer learns your store once, and every screen that behaves differently costs them.

The photograph is usually the product page. Everything else is supporting it.

Where a mobile app comes into it

The same screen designed several ways is the point: a catalogue of four hundred lines needs a different home screen from one of forty.