What are the consequences if I allow a Bard to Inspire himself?

Watching a video stream of a game, one of the players, the Bard, asked if he could inspire himself with his Bardic Inspiration feature. The GM objected and said that the feature specifically said that you must target another target with your Bardic Inspiration. I thought this was strange and when I looked it up, he was right. In the end, the GM let the Bard inspire himself and he used the Inspiration die to succeed on an ability check.

I find it weird that Bards can’t inspire themselves and I can’t find a mechanical reason why they should not be able to; they are still expending a resource, anyway. Thankfully this hasn’t come up in our games as the Bard is perfectly happy inspiring others while playing a folk metal tune on his phone. But if it does come up, I’m inclined to say why not?

Is there a mechanical consequence I’m potentially missing if I allow a Bard to Inspire himself?

Would that break the class?