Dealers quote clean prices because they want a number that reflects credit and rates, not the calendar. If quotes included accrued-interest, every bond's price would creep up for six months and then drop on the coupon date, which would make charts useless.
Screens, indices and yield calculations all use clean prices. The actual cash that changes hands is the dirty-price.
Example: a bond is quoted at 101.25. Accrued interest is 0.83 per 100 of face. On $250,000 face you pay (101.25 + 0.83)% x $250,000 = $255,200, even though the screen says 101.25.
Related: dirty-price, accrued-interest, par-value, bond