Leave is a unilateral action which can be implemented full stop.
True, but - under the UK system, parliament is in charge. In order to implement anything, you need to get parliament to agree on it. If they can't do that, then it doesn't get implemented.
By saying "Just get on with it", you're effectively asking parliament to forget about its own rules - heck, even its own existence - in favour of the outcome you want. You are too impatient to wait for parliament to do its job. Really, you're very close to saying that parliament is not fit for purpose.
But if that's the case, then you're on even shakier ground, because it was parliament that called the referendum in the first place. And if they have no legitimacy to make these decisions, then the referendum itself was illegitimate. You can't have it both ways.
"Democracy" is not magic, it doesn't always make decisions clearly, it doesn't always make them right, and it doesn't force anyone to commit to a bad choice. That's your decision, or rather it's the decision of each individual MP and of parliament collectively. Anyone who hides behind the referendum result, claiming that their hands are tied by it, is debating in the worst kind of bad faith.