Choosing the correct form of a verb

Since "the key to" is an idiom meaning "The crucial element in," it follows specific grammar: as noted briefly in a comment here, the phrase is followed by a noun or gerund, sometimes with an adjective before it: the key to long life, the key to a happy marriage, the key to enjoying your retirement, etc.

Using "choose" would fail to recognize the idiom.

The headline is making use of a pun on the word "key".

A key is part of a door with a lock, and you have to choose this wisely. So you would use the verb "choose" in this sense.

But "key" can also be used more figuratively to mean something that provides a solution to a problem. In this sense, the problem is choosing a door handle and lock, so you use the gerund form "choosing" to treat the process as a noun.

The title uses the latter form because that's what the text is about. They chose the phrase "key to" just for the clever word play; this is common in headlines.

The difference is between a to-infinitival as a relative clause, and a gerund-participial complement to preposition to.

As a relative clause, the key could be subject and take agent role in the situation:

the key to choose wisely = the key which chooses wisely

?the key chooses wisely

It would seem a bit odd to put an inanimate object in the agent role, but it does work sometimes:

Education is the key to get them out of poverty.

=

Education gets them out of poverty

the key could also be understood as object, with the subject left understood

the key (for you) to choose ___ wisely

You choose the key wisely

This seems a bit awkward with key, but does work just fine with other words:

the decision (for you) to make now is whether to give them the money

you make the decision

Contrast this with:

Education is the key to (us) getting them out of poverty.

=

We get them out of poverty.

Here, with to separating key from getting it is clear that key is neither subject nor object in the clause headed by getting, but rather the means by which this is done.

The clear choice for the advertisement is this last construction as the key is neither doing the choosing nor the thing being chosen, but the means by which one may choose wisely.