I guess the best way to look at it would be an example like this:
The Bard and party are fighting. The Bard looks over to the Warrior who is facing a 2 vs 1.
The Bard speaks, "Don't worry, Jerry! You can do this, you got this, bro!"
Jerry feels inspired by the Bard's words of encouragement.
Jerry can now add 1d6 to any 1 attack roll or saving throw within the next 10 minutes.
I still feel like the Bard should have a multi-person one, even if it's a die-grade lower, and a 1 minute duration.
Like for example if the Fighter, Rogue, and Wizard are all fighting multiple opponents and are near each other.
The Bard speaks, "Don't worry, Ted, Jerry, and Vanessa, we got this! We've faced far worse before and have come out on top!"
Ted, Jerry, and Vanessa can now add 1d4 to any 1 attack roll or saving throw within the next minute.
But...
1 - the bard does have that ability - it just takes more actions. What you are asking about is the ability to use all at once losing only one bonus action. A quick look thru all the various things bonus actions can be used for shows the gains here are more than just the spread of the dice but the gain over the combat of two-four bonus action uses in cases where you wait until combat to use it.
"hey, its the big boss, not just another skirmish so... i use my bonus action this turn and everybody gets a inspire die" vs "...i use my bonus action for the next 5 rounds to give bonus die" is a massive difference, considering both the delay in some getting their inspire dice and your own loss of bonus actions over the next 4 rounds.
See, by keeping the attention on the dice and the gain, you are distracting from the real impact and improvement: getting a multi-target spread and the massive gains in the action economy and timing.
i mean, hopw often is it that the first couple of rounds are where some of the biggest effects get to flying? how often is it that for a combat where it matters you don't need to use a one-use bonus in the first 10 rounds but need it in the next 10 minutes?
2 - With the decision made after roll but before result - the difference between a d4 and a d6 is rather trivial. On any given combo of roll vs DC vs circumstance... it can be huge or not but as a general bonus in the player's control - the net result of getting everybody a d4 option from round one is YUGE compared to the gaining of a d6 one by one over three rounds.
Its basically a change which says "bards get extra bonus action uses" for any case where they would want to help more than one person combined with "bards get to cover everyone more quickly" in similar circumstances.
i bet most any fighter in the game would take "drop d10 to d8 for your second wind uses but you can use them all one one bonus action."
i doubt i would even allow this big a gain in actions and yield for a class already performing at par at least even with a feat.
Would you allow a caster to take a feat which said "on one turn you can cast as many spells regardless of actions up to your casting bonus if tye use a spell slot higher without the normal up-slot gain"?
Thats not just a shift in cosmetic sense but a serious nova potential and a serious edge for the "boss fifhgt" outputs.
to me, allowing a one-use all-expended option shifts the bardic inspiration heavily into a "best held for one big fight bursty" thing in a YUGE way (saves multiple bonus actions) while the requiring bonus action for each gives you more of a "use it along the way" feel which lets the bard inspire be more of an ongoing thing - not a nova must for the first round of a boss fight.