Is a preposition of place required when replacing "where" with "that"?

Question

A preposition appears to be needed when that is used in sentences such as:

That's the store where I bought my computer.
That's the store that I bought my computer (at?).

with exceptions occurring, per page 483 of Oxford Practical English Usage, for "somewhere, anywhere, everywhere, nowhere, and place."

That's the place where I bought my computer.
That's the place that I bought my computer (at?).

A comment on a related question at ELL states:

In practice, most native speakers wouldn't get bogged down in prepositions and relative pronouns here - we'd just say "We need a place to stay". But for no specific reason that I can identify, if I replace place with house there it seems to require the preposition: "We need a house to stay in".

I agree with the notion that the preposition is required following that. What I can't do is wrap my head around exactly why that should be.

What difference is there between where and that in these sentences that calls for the omission of a preposition when using where and the inclusion of a preposition when using that?

Answer 1

Perhaps it will help to transform the clauses to their equivalents that better show the syntax. "That's the store where I bought my computer." becomes

Where did I buy my computer? or
I bought my computer there.

No preposition of place is required with words like "where" and "there" that contain the notion of a place in their meaning. "That's the store that I bought my computer" becomes

I bought my computer that

which doesn't work with "that" serving as a conjunction with no other role in the clause. This works if the clause is a direct object:

I didn't steal it. Did you ever consider that I bought my computer?

but not if you want to modify "store."

You'll have to consult a linguist to answer the question of "place." Perhaps it sufficiently names a location such that no preposition is required. The OED has usages with "find," "have," "hold," "give," "make," and "take" that don't require a preposition, some of them going back to the 14th century, so perhaps we're just used to idiomatic expressions that contain the word without a preposition.