There is one huge problem with this, which is that under no circumstances can you reward people for placing larger orders than they can actually execute. If you do, you encourage people to make large orders in the hope that only part of them will execute, which can have catastrophic consequences if for some reason the whole trade executes in some unexpected case.
Any form of lottery is vulnerable to this. If you clamp all the timestamps to ten minute intervals, then you need some arbitrary rule to decide which of the valid trades in some time window to execute. If you randomly choose a few, you encourage people to put in more orders than they can execute. If you satisfy them partially, you encourage people to put in larger orders than they can execute.
As a result, the only way to do this that encourages proper behavior is to have some arbitrary competitive rule in place. The easiest and most obvious such arbitrary rule is the fastest order wins. This might seem anti-competitive at first glance, because it keeps out the tiny investors who can't get enough capital to compete in this area. But really, it's not that bad because it only takes a few million dollars if that to get top-notch hardware in a Manhattan or Chicago datacenter. In addition, market-making is by-and-large a commodity service: the entire HFT community can only make money when the market is inefficient, and market-maker competition makes the market more efficient. The HFT world makes money off of market inefficiencies, which are capped by the size of the market and the amount of outside investment. It's a hyper-competitive world of arbitrary rules, in place to encourage a more efficient market for investors. So you don't want to do it yourself, and you don't want to regulate them because you want HFT to be a insane low-margin nightmare for the benefit of everyone else.