Is the preposition optional in "going down (to) the store"?

Cambridge Dictionary says: In informal situations, we can use down to talk about a quick trip to a destination which we consider to be less central than where we are. In this meaning, we can use it with or without to. Without to is less formal:

I’m just going down (to) the shop. Do you want anything?

Are you going down (to) the golf club tonight?

Is Cambridge wrong here?

I haven't ever heard "I am going down the shop" unless used to say "I'm perusing the aisles of a store", and not "I am going to the shop."

Maybe it's different in the UK? I can't even see how this would begin to be right.

I understand that

• I am going down the store

and

• I am going down to the store

are two different things. Is there a dialect / is it "informal" to mix those two phrases into the 2nd definition?

The Longman Dictionary Online confirms the informal use when omitting "to" in such expressions

go down the shops/club/park etc British English spoken informal to go to the shops, a club etc

Does anyone want to go down the pub tonight?

At risk of sounding classist, elitist or snobbish, I'd say that the expression is mainly used by the working class and younger people.

A British band called Sham 69 has a song named "Hurry Up Harry".

The lyrics include

We're going down the pub

Come on come on

Hurry up Harry come on

Come on come on

Hurry up Harry come on

We're going down the pub

We're going

It's better down the pub, a national campaign celebrating Britain's passion for pubs was launched in May 2013.

As far as I know, the word "shop" is also mainly used by British English speakers as opposed to saying "store".

This EL&U answer as well as this one cover the meanings and etymology of "shop" and its variations.