"Regular" transistors? Hmmm...
MOSFETs have their power issues as well. Their driving source is voltage into a (normally) hi-Z node, and static power is ideally zero, but simple CMOS logic shorts the rails every time the gate changes state so the power goes up with frequency and voltage (squared): P = k* C * V^2 * f where k is a constant, C is the effective capacitance, V is the voltage across the gate, and f is the switching frequency. The crossover in power used to be around a few hundred MHz for roughly equivalent BJTs and MOSFETs (do not know if that is still true). In The Olden Days leakage current, the current drawn by a MOSFET when it is not switching, was minor and usually neglected. These days, for the tiny little devices we use, leakage is often half the power of a large chip, and that is power wasted whenever voltage is applied to the device. Leakage scales linearly to exponentially with voltage and temperature (it's complicated) and as devices shrink, even though voltage drops, leakage increases. However, MOSFETs are generally easier to make (fewer processing steps), and can be made very small, so they dominate logic circuits if not analog (at this point most of the analog is probably part of a digital chip). BJTs tend to be bigger overall and require larger isolation regions around them. Sometimes.
Now we are working with FinFETs and looking toward The Next Big Thing.
Personally, I like Ron's duct tape idea...
