Busy is often a coverup. You know it. Admit it. Go ahead. You know.
Busy provides you the side-step-away option you're looking for. It allows you to dodgeball out of the way from what you know is needing direct attention but . . . maybe just too touchy or complicated to face, address and deal appropriately with.
Who wants to open a can of worms? Or knock about a hornet's nest?
Best to leave the could be, might be, probably will be sensitive people facing issues alone, even though we know the hide-your-head-in-the-sand approach to responsible management is anything but . . . responsible management.
Real leaders focus on issues. They don't get all self wrapped up in dramas. Italian operatics are fine on stage, if you like that sort of show, but its not something to promote and stage at work with the cast of characters you're responsible for.
To deal with sticky, tricky, sensitive people issues - focus on the business issue not the personality. Return to the business case for why and how, the role in question needs to perform for success across others, then gap your person to what the role needs to succeed.
When you're ready to see the power of leveraging behavioral and emotional intelligence to help your people win more, win better and win more smoothly, you'll find the time and take the time needed to develop your own self-awareness and to support others developing theirs so they can do what's needed to win where they are.
Time here isn't the issue. Nope. That's the code, the code for "I'd rather not deal with it, and its easier to blame it on being OMG, so very busy!
As James Humes said, "The art of communication is the language of leadership."
Do you have time to be a leader. You may fool yourself and say you do. But you can't fool others. How you show up is way more transparent to others than you think.