Difference between "on my way to be 25" and "on my way to being 25"

Close to and (the) way to both take -ing clauses, not infinitive clause.

There is no logic or general rule to this, and it has no discernable relationship to the meaning: it is just a property of those particular words. It is no more predictable than the fact that the adjective close cannot mean "which has been closed", whereas the adjective open can mean "which has been opened".

For examples of similar words that go the other way, consider ready to and prepared to, both of which require an infinitive clause, not an -ing clause.

The World Trade Center is, famously, the complex of buildings destroyed in the 9/11 attacks and now rebuilt. However, you could say China is close to becoming the centre/center of world trade.

We would not use the infinitive in sentences like this.