When they decided to stop minting pennies I think they should have gotten rid of nickels and (I know this will be controversial) quarters as well!
Keep dimes and ramp up production of half dollars. Then we can just drop the second decimal place and standardize pricing everything in 0.1 dollar increments.
The fact that quarters are still somewhat commonly used in machines (vending machines, parking meters, laundry) is probably the biggest practical obstacle.
I would have gotten rid of nickels and dimes; then everything is priced in 1/4 dollars.
This may be the most practical go-forward plan. The Euro's .20 coins are also attractive too. But you're correct that quarters, as the smallest common currency that you can plausibly buy something with just a couple of them, are just everywhere, from laundry to car washes, so the pain in retiring them would be widely felt.
What I've learned from the penny retirement is that people are deeply distrustful of simple high school level statistics! Millions of people have angrily seethed that somehow stores are or will be using the penny retirement to rob them, despite knowing that most transactions have an unknowable amount of different items, and sales tax, so attempting to manipulate prices to gain a statistical advantage out of rounding would be incredibly difficult and would yield a pitiful return. Let alone how the cash transaction share is declining every year.