Is "looking to" acceptable English in this use?

From the American Heritage Dictionary:

Usage Note: The phrasal verb look to has recently developed the meanings "expect to" and "hope to," as in The executives look to increase sales once the economy improves or I'm looking to sell my car in July. In a recent survey, the Usage Panel was divided almost evenly on this usage, with 52 percent of the Panelists finding it acceptable and 48 percent rejecting it. Of those rejecting this usage, a small number volunteered that they would find it acceptable in informal speech, and in fact the divided response of the Panel may be due in part to the informal flavor of this phrase.

I think this may answer all of your questions except "Why is it recently popular". It may be popular because in my opinion, it doesn't quite mean either "expecting to", "planning to", or "hoping to", but somewhere in between; less definite than "expecting to" or "planning to", but more definite than "hoping to". So it may fill a niche which, until this phrase, had no good verb to fill it.

To be looking to do something is to plan to do something. You can use

Jeff is planning to start something big

as an alternative.

The Recency Illusion strikes again. Look followed by to followed by an infinitive has been in use, in the sense of ‘expect’, since Thomas Hobbes used the construction in his ‘Leviathan’ in 1651.

I mainly encounter "is looking to", in a British space (regulatory affairs), as meaning "intends to" where the subject is a person, otherwise "is intended to". Example (not written by me, as I dislike this construction) "The Targeted Charging Review is looking to implement changes to the way demand users pay". That clearly should have been written as "The Targeted Charging Review is INTENDED (or, has been designed) to implement changes to the way demand users pay ..."