Compound is important but not so much as the amount of movement the tread pattern allows, which is where the heat is generated from. A good quality road tyre will still have a decent level of natural silica in its make up that provides an ability to generate heat, alongside more tread blocks that move and therefore do warm up quicker than a semi slick. This is where on a track the road tyre loses out - the heat level is reached quicker, but the movement in the tread blocks creates more slip, meaning the tyre will have less outright grip at the extreme track use end of the spectrum. My experience of the Falken however is that it guides you to this point really clearly, allowing you to decide just how much 'slip' you want to initiate!
When driving on road, it's useless to call a tyre a "tin top" tyre or say that something is designed for much heavier cars unless you know the make up of the sidewalls, the layup of the tyre and the materials that go into it. For example, I ran Kumho V700 tyres when sprinting, which were designed for rally cars weighing twice as much as our cars, so the sidewalls are super stiff. However, that means you can run much lower pressures to create the movement that generates heat. They did alright for me and several others!
To answer this question more simply:
Not necessarily so - winter tyres have additional sipes (cuts through the tread blocks) deliberately added to increase tread block movement and generate more heat, and grip, at lower temperatures. A winter tyres provides 25-35% more grip than an all round tyre in temperatures below 7c, and especially in snow or icey conditions. It is also less about the label of the compound and more about whether the 'rubber' in the tyre is synthesized (such as in cheap ditchfinders - it's why they last forever but have zero grip) or natural (such as expensive Avon slicks, for example).
To challenge the thought process further - what is faster in the wet on an F1 car, a full slick or a full wet tyre? Sure, clearing the water is part of it, but not the whole story. Answering that might also help you decide whether a good road tyre or a cheaper semi slick is better as a road only option.