why is this sentence correct?

[zahari]

hello everybody!!!! i would like to make a question. i have a sentence:

The people________there first,will get the best bargains.
a. getting
b.that get

the teacher's book gives "b. that get" as the correct answer. however,i know that a present participle can take the place of a relative clause.why can't it be the case here? is it because the pr.participle cannot take the place of a defining relative clause???

[Casiopea]

Welcome, zahari. hello

Compare,

EX: People (who are) arriving at Gate 15 will have to go through customs.
EX: ?People (who are) getting there first will get the best bargains.

The first event is happening now; it's actualized, whereas the second event is unactualized. That is, our example sentence is conditional:

EX: People will get the best bargains only if they get there first.

Now actualized; i.e., it's happening at the moment,

EX: People (who are) entering the store right now will get the best bargains.

Correction
I would like to ask a question.

[Belly T]

I still don't understand your answer,Casiopea.
I considered the two sentence above are all correct.
Can you give me more definition?