If you start to drive your TX path with "small" (depending on the number of channels) FPGA or CPLD, yeah this is more possible than with a raspberry pi GPIOs, this is even how some good ultrasound machines are build (depending on the volume)
In the end the cost of the hardware will be not far from the production cost of a "conventional" ultrasound machine. Because that industry has "low" volumes and very high R&D costs (especially for machines intended for diagnostic use), the difference between production cost and market price is very high. If you can eliminate part of that difference because you operate in another context, that's one way to drive the price down.
So yes, it should be possible to build one with limited capabilities for a few k$ per unit. (Spending various amount of money on R&D, depending what you're doing precisely.)