While your question is pretty thorough, I'm assuming a few things in this answer that you didn't clarify;
When you crank the engine, and it doesn't start, I'm assuming it's a smooth crank without hiccups. By that I mean it isn't trying to start, it just cranks.
You have the petrol model, not a diesel.
It uses an electronic push-button start mechanism.
From what you describe, it sounds like either an ignition fault, or a fuel-supply issue. Since the car runs when push-started, and since modern vehicles are fitted with electronic timing systems, it's unlikely to be that, so most probably a fuel-supply issue. Maybe the pre-start fuel pump is on the blink?
A Test You Could Try
Some Renault's have a manual primer pump that looks like a black rubber bladder under the bonnet. Your user manual will give more details - it's usually used in the event that you run out of fuel.
Try pumping it a few times before you start the vehicle and see if that helps - if it does then you probably need a new fuel pump. Best to confirm with a garage. If you don't have that device, you can't test it unfortunately.
If you park on a steep gradient and there isn't tons of fuel in the tank, it might not be picking up any, but that's very unlikely in modern vehicles (used to be a problem in older Kia's).
If the spark plugs are clogged or there's loads of water around the ignition system, it might be difficult to start when cold. However, if it's regularly serviced there's no reason to suspect either.
That's all I can think of right now. I hope you figure out what's up!