Hmm, I forget if BJTs are scale limited.
FYI, electrons and holes "flow" just fine on their own, without any applied force: diffusion allows them to spread out without any motive force at all. Indeed, this is the dominant means of current flow in BJTs, and the reason they work so well!
You do tend to have the problem, with BJTs, that, specifically because of diffusion, you have to carefully isolate them from each other. Otherwise one transistor conducting nearby will release charge carriers into the silicon substrate and anything touching -- the diffusion length is several microns, so a sensible leakage current can be found at several transistors' width away, even for bulky ~2um construction typical of the 70s (e.g., LM741, etc.).
I believe those problems were usually solved by making a N-well within a P-substrate; this forms the collector, and the collector-substrate junction shunts the leakage (as collector current proper!), keeping it away from others. Subsequent P and N diffusions formed the base and emitter (of increasing doping level, as necessary to more than overcome the existing doping from previous steps; which is why the B-E junction usually has a very low breakdown voltage).
MOSFETs nowadays still have to deal with similar problems; this was solved by etching deep vertical gaps between transistors, so that the diffusion path must be vertical as well (down, into the substrate, over, and back up). Such a solution would work just as well with BJTs.
BJTs haven't at all disappeared; with the introduction of strained SiGe:C processes, HBTs (heterojunction BJTs -- even better than the regular kind) have been used in ICs in great quantity already. The higher fT/Ic figure means higher speed devices on the cutting edge, or even further reduced current consumption for conventional-spec parts.
Tim