There is a grace period - you get a whole week to assign the bounty. How would adding a 3 day "grace period" be any different from just making bounties last 10 days? You can assign the bounty any time after the first two days, so there's if there's an answer you like, assign the bounty a little sooner. Unless I'm missing your point, I feel like this is already built into the system.
Furthermore, if you haven't assigned a bounty and there are 3 days remaning, you get a dropdown notification bar the next time you visit the site. When there are 24 hours remaning, you get the following message in your email:
#Reminder: Your Bounty Is Ending Soon!
Reminder: Your Bounty Is Ending Soon!
Your bounty on the question …
Question name and link
… ends in 23 hours.
Don't forget to review the answers and award your bounty by clicking the bounty indicator to the left of each answer. If you don't award your bounty, the highest scored answer (with a minimum of 2 score) provided after your bounty started may be automatically awarded half the bounty amount!
If that's not a clear enough indicator that the bounty period is ending, I don't know what is.