To be clear, the FTC is warning all online influencers and not specifically those who receive free board games to review. It'd be wild if the FTC thought the problem of board game reviewers not disclosing this was big enough and egregious enough to warrant special action against them.
Of all the actual misaligned incentives out there, giving someone a free board game in exchange for them playing it and creating a review seems like a great thing. Someone gave you a $50 game and you created a free $500+ video about it encouraging others to take a break from screen time. I hope no one tells the FTC about paid Amazon reviews, they'd probably have a heart attack.
Sure.
But if you don't reveal that they've given payment in the form of a free game, you are giving a dishonest review. Free stuff is payment. And everyone knows it is payment - Free meals and other benefits at a job are advertised to entice you. It might not be taxed, but is definitely part of your pay package.
I'm pretty sure the FTC knows about the Amazon reviews - they've taken action against some of them. They may or may not have the budget to go after more and may or may not have the power to go after all of them.