Yours is a good thought too. Here is my perspective/experience:
Scenario 1: You start at the high end of the range or just a bit above it, I think you risk the employer coming in low as a counterplay, expecting to meet in the middle. Now, it's up to you whether you do actually meet in the middle, or hold firm and move them way up, or just end the negotiation entirely. But I think this opens up their strong counterplay and establishes their ability to take the reasonable-seeming ground that it's fair to meet in the middle (even if it isn't).
Scenario 2: They throw out the first price.
Scenario 2a: They throw out the first price, and it's low for the range. Now you can counterplay by going high. You also can take the strong moral high ground here by showing comp data and basically saying that it seems like they're not being fair or don't value your experience and skills correctly, which is a concern to you. Often, this will be enough to get them to offer a higher price, and you can then spring from there to continue negotiations from this higher spot. Whether they offer a higher price or not, you can start high and employ meet-in-the-middle even from the low starting point. So, Scenario 2a is no worse than Scenario 1, and has some chance of being better.
Scenario 2b: They throw out the first price, and it's mid- or high for the range. Hey! Free money! Now you can safely shoot higher and hopefully end up somewhere between where they started (already on the good side of the range) and where you countered. :-D So, again, Scenario 2b is at least as good as Scenario 1. Very rarely will a company demand no negotiation whatsoever (and maybe you want to tell them kiss off if they do, unless they have a very good philosophical / values-based reason for doing this, and they do it consistently with all candidates), so you're only bettering your position by opening the possibility for them to start somewhere above the low-end of the established market range.
By combining all the above logic, you don't want to go first.
In practice, I think this works a lot better for prospective employees too. :-) I personally think companies should offer first, for this reason (even though it's bad "negotiation" on the company's part... I actually don't think employing people is a zero-sum game, let alone a one-turn game... so I think you're better off trying to position them well. But of course not all employers think this way, and it seems in fact most do not.)