Why Do People Exchange the Purchase First, Then the Money?

Whether trading an asset or service, as in last week’s Parsha:

ותאמר לה בת פרעה היליכי את הילד הזה והינקהו לי ואני אתן את שכרך ותקח האשה הילד ותניקהו.

I asked a friend and he said this was because everyone already has money, so in the case of a dispute, things are less obvious.

Here’s my own answer:

In a barter transaction, each person simply prefers the other item. But money is presumably objectively more valuable to all parties. Otherwise, it wouldn’t be money in the first place. Money is the medium of exchange because it is acceptable to virtually everyone.

What do you think?