This is an interesting idea, but I'm afraid it will need some refining, even beyond transistor Vbe (Vo).
Let's start with Vbe, though. Unfortunately here, Vbe is a property of the semiconductor materials used, which means we can't set it arbitrarily. The 0.7V value you quoted wasn't some arbitrary standard we chose; it's what you'll find on any silicon BJT. Germanium transistors - the other popular variety of which I'm aware - have Vbe even lower. If someone knows of significantly higher Vbe's, please let me know in the comments so I can add them here for completeness.
There's a more fundamental problem, however. Here in your idealized circuit, RL is in parallel with your transistor, and V1 connects to both sides of RL. This means that your transistor will have no effect, whatsoever, on what RL experiences. To see this, just apply Ohm's law for RL. The current IL = V1/RL, no matter what's going on in other parallel branches.
Your transistor, however, will indeed draw more current as V1 increases. Assuming a beta of 100, your current into the BJT collector will be 100*((V1-0.7)/(1kOhm)). This will heat up the transistor, which causes it to draw more current (since it's a BJT). I'm not sure if the conditions are right here for thermal runaway, but I'd definitely want to think about it pretty hard before I wired it up to a car battery. And yes, as you noted, it will hit a point where the transistor is drawing more current than it's capable of, and the air will be filled with the lovely aroma of magic blue smoke.
But hey, that's the smell of learning! I've practically burned enough transistors that I could have recycled them into my diploma. :)
Note: There is a circuit, conceptually similar to yours, that could work.
simulate this circuit – Schematic created using CircuitLab
The Zener Diode will drain any voltage above its set voltage, keeping the voltage across RL from exceeding that set point. Of course, by Ohm's law, this means that the current through RL is also limited. It's not the most energy-efficient way to accomplish this, but variations on this theme are common where price and/or size constraints are more important.
The Zener diode doesn't burn, since R1 is in place to limit the total possible current through V1 to I1 = V1/R1. That would hold even if RL or the voltage regulator failed short.
There are also ways to do this with voltage regulators, but as your question wasn't really about alternative ways of accomplishing this feat, I'll stop here for now.