Mainly for strength. Anchoring the strings under the top (with or without bridge pins) means there is stress in one small area that is more than the top itself (or the top plus a conventional bridgeplate) can handle.
You can shift some of this structure to the underneath side of the top, in essence producing a two-piece bridge.
While trimming some off the lower half of the bridge is possible, there is a point where you are compromising the ability for the bridge to hold the saddle upright.
Low mass (hollowed out) bridges have been tried, with mixed results. As the bridge weight is reduced (particularly as a proportion of the total top weight), the guitar tends to sound more and more like a banjo, meaning louder but with less sustain and less pleasing tone.
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