Probably the reason for using mechanical shutters is that their disadvantages are easiest to live with; competing technologies are not (yet) clearly superior.
The major problem is that electronic shutter affecting whole sensor at once is rather easy to be implemented on CCD sensor, while for CMOS (preferred on new DSLRs) it requires additional circuitry in each sensel. This can be done, but the expense is high - dynamic range, resolution and/or cost. For example, Sony F55 is a video-oriented camera body with full-frame 8.9MP CMOS sensor having the global shutter circuitry, and costs whopping $29k in 2015 - about 4 times more than the top-end bodies from Canon/Nikon with much higher resolution but without that shutter circuitry.
Normally, CMOS sensors are reset and read row by row, which takes more time than it takes a modern mechanical shutter to travel, therefore rolling shutter effect is worse and max sync speed is slower.
Many recent cameras provide option of electronic first shutter curtain, where exposure starts by sequentially enabled sensor rows and ends by mechanical shutter curtain following at the same speed. Current sensors can reach from one edge to another a little slower than 1/100 seconds, on par with mechanical shutters from 1970s. Although faster speeds are possible by exposing only a slit at a time, this speed determines amount of rolling effect and max sync speed.
For fully electronic shutter, the rear curtain has to be electronic also. This will turn off and erase the row, so data has to be read out first. Reading data is even slower than turning the rows on, crippling max sync speed and intensifying rolling shutter effects a few times more.
In video/live-view mode, the electronic shutter can be "sped up" by skipping most rows, resulting in a lower resolution. Transferred data can be reduced even further by decreasing bit depth - this manifests in reduced dynamic range.
CCD is commonplace in compacts, and they often use electronic shutters. There have been electronic shutter CCDs used for higher speeds on some older Nikon DSLRs, like D1 or D70. On these cameras, grid-like patterns were reported to sometimes appear on plain tonal areas with shutter speeds that used electronic shutter.
I suspect you only had the commonly used focal plane shutter in mind; leaf shutter is another mechanical design for a shutter. Its main benefits are quietness and ability to sync flash at any speed, because the shutter always fully opens. But leaf shutter either needs to be located just where aperture diaphragm is (i.e. in every lens), or needs specially designed lenses that have a nodal point at certain distance between lens and image plane. The first option is expensive, used in many medium format systems; the other is restrictive for lens design, but has been used in some old SLR models (e.g. Topcon Auto 100).
On the Canon SX30IS, the shutter (which can go at 1/3200s) is most likely not the limiting factor for burst speed. The speed is more likely inhibited by data bandwidth - even when you shoot low-res, camera still reads all the 14MP from sensor to give maximum image quality. In video, image quality is less important than frame rate, so the camera reads only selected rows and columns off the sensor.
According to specifications, turning off LCD should help you reach 1.3 fps. Or if you'd like to trade image quality for burst speed, just shoot video and extract frames later.
