Why is that the local police are not permitted to fire an arm unless the situation is reasonably dangerous? This is reinforced and practised under an honour system and even a legal system to an extent. I'm sure other things like that in the Muggle world are. There is certainly no way to detect underage magic, but it is restricted under the same honour system - parents are expected to see that the rule is followed.
It all seems to operate on a sort of honor system, but there are plenty of wizarding folk without that particular virtue.
But can a person use them at the same time? We don't appreciate the complexity that may be the wizarding world, we under-estimate their laws. Like the issue of not being able to create food out of thin air - yeah nobody will be hungry if they continued to increase the quantity of what they have but question is if somebody is poor and has only bread then how much bread would he want to consume? Perhaps we ought to be saying something along the possibility of someone being not well-fed not that they can't go without food at all.
Here's something to consider, what if you used the potion and the time turner at the same time? Talk about perfect possibilities.
In my opinion the Ministry should be given a little more credit. It's the place the regulates everything related to magic there is. I'm not sure time-turners are readily available to anyone who asks. As others mentioned the potion is a complex one and we can easily assume not many people can make it. Not many people in the wizarding world have the resources to have both the things at the same time.
Crouch Jnr uses the Polyjuice potion but not Felix Felicis. Dumbledore never uses though I'm sure if he did it'd have been a cataylst to finding the horcrux(es). Snape isn't overcome by the temptation to make one in Harry's third year - I'm sure he was absolutely dying to lay his hands on Sirius. Lucius must be desperate after things started go bad for him with Voldemort to get some luck. But it's use is not prevalent. Slughorn is a different class of wizard. To him like Ollivander, power and control are at the same time both frightening and impressive notions. To characters like Crouch Jnr, Snape, Dumbledore it is not a matter of availability. These people are masters of their craft, they would rather use their knowledge, cunning, manipulation to get things done rather than brew a potion. It's just how intellectual men would operate regardless of what resources they have at their disposal.
But here's a thought: couldn't Crouch/Moody just take the felix potion himself, ensuring that his evil plans come together and succeed?