Bards are meant to be social characters that buff the party. Having a bard in your group is wonderful, because it means your rogue can be that much sneakier, your fighter can lift that much more weight, and your monk can do that many more flips in mid-air when they’re inspired by their bard. It builds party cohesion and it makes the other players feel really cool.
Because the bard is not able to give herself inspiration, she is necessarily going to spend the majority of her out-of-combat time buffing the party. People love being buffed! And with bounded accuracy being such a huge presence in 5E, the addition of a d6(3.5), d8(4.5), or d10(5.5) to a skill check is huge! It’s massive! It’s amazing!
…Until you realize that it is huge, massive and amazing for anyone. In 5E, the world’s strongest, most athletic barbarian has a pretty decent chance of losing when arm-wrestling anyone off the street. A monk who has spent her entire life training to be an acrobat can expect to lose a tumbling contest against the local tavern owner, and when asked to do even trivial tasks like climb a ladder, she will fall off about 10%-15% of the time. A 9 foot tall, battle-scarred fighter who has fought literal gods can very easily fail to intimidate a small child. Bardic inspiration can make huge differences in these checks. When you add that to the fact that the bard is already an extremely skilled class (Jack of all trades and expertise skills), you’re giving the bard less and less reason to waste his precious bardic inspiration die on the unskilled losers in the rest of the party, and more and more reason to buff themselves to the heavens.
If your party bard is very selfless, this rule change will likely not affect the game. However, if this bard has any amount of selfishness, there is a great potential for them to spend the bardic inspiration dice solely on themselves. In fact, when your bard gets Cutting Words, they will be able to make checks with a +d6/+d8/+d10 themselves, AND a -d6/-d8/-d10 to other people (including party members), meaning a midlevel bard can expect to have a 9 skill point advantage on any check they want. Someone with that kind of power will be able to easily make themselves party leader and essentially go unchallenged in any and all skills. That same bard can lie to the entire party and virtually never be found out, pickpocket anything from any party member with no chance of being caught, and intimidate even the most seasoned barbarian in their sleep.
If you don’t foresee your bard doing this then you’re really just giving him a high-level bard ability for free. If you do see your bard having a bit of a devious streak, out-of-combat encounters are going to become ‘The Bard Show’ and the rest of your party will likely just sit around, waiting for combat while your bard raises armies, topples kingdoms, and charms great swaths of people with ease, all while out-acrobating the monk, out-lifting the fighter, and out-perceiving the druid.