I would like to sell a handcrafted wooden outdoor item, and paint my own NFL or college team logo on the item to sell locally (flea market or festival).
I agree with my colleague, no this isn't legal. Using an NFL's or colleage team's trademark is trademark infringement, and wrongfully trades on their rights. This is true whether or not you'd be selling the same products or those that are closely related to those they they sell. Doing this would unfairly compete with and dilute the TM on the products made or licensed by true owner, and consumers would be justificably confused and misled about whether your company is the offical source of the products, which you're not.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Patent Infringement Attorney
Using any team's protected logo or name could be confusingly similar to third party purchasers and infringe the rights owned by the team, NFL Properties or the NCAA. The type of use you describe would wrongfully trade on the owner's rights and be actionable by the trademark owner if not properly licensed. You may want to contact the appropriate trademark owner and seek a license to avoid infringement. An intellectual property attorney can assist you in negotiating a license.
Good luck in your business.
Disclaimer: This answer does not establish an attorney-client relationship and does not constitute legal advice. It is for general information purposes only.