Why the clean price is mostly quoted in the US bond markets and the dirty price is mostly quoted in the European bond markets?
The premise of your question is wrong. European bond markets usually quote clean prices (without the accrued) for performing bonds - exactly like U.S., Canadian, and most Latin American bond markets.
Both in American and in European bond markets, bonds begin to be quoted dirty (with the accrued, total proceeds) only when they are on the verge of defaulting.
However there are examples of other bond markets where the market convention for performing bonds is dirty quotes: Brazil, Argentina (even USD-denominated local-law bonds are quited dirty), et al.
The main reason why most markets choose to quote clean price is simply that plotting the dirty price of a performing bond against time displays an annoying zig-zag pattern, rising as the coupon accrues, then jumping down on every coupon's ex-date. You can see these zig-zags if you plot against time the price of preferred equity or dividend-paying common equity, that similarly jump down on every dividend's ex-date. The zig-zags go away if you convert the price series into yield or a spread over treasury benchmark or Z-spread etc. However not seeing right away how much the observable price changed just from accruing another day's worth of carry, rather than from something moving in the markets, for example, when you look at Brazil bonds price series, is mildly annoying.