Today I consciously realised something that I have unconsciously known since primary school at the very latest, and that is that while ‘take a look’ basically means ‘look’, ‘take a seat’ doesn’t mean ‘seat’ – it means ‘sit’. This is because look is both a noun and verb, while seat is a noun (usually) and sit is a verb. The longer form with ‘take a’ has to be followed by a noun (I’m kicking myself now how obvious that is), while the short form by itself has to be a verb.
There are subtle differences in usage. Saying ‘After work this afternoon, I walked’ sounds strange, while ‘After work this afternoon, I took a walk’ sounds usual/natural. (Following it with ‘the dog/to the station/(for) one hour/(for) five kilometres’ all add to the possibilities.) Being asked to ‘Sit’ would sound far too abrupt (‘Sit, please’ is just possible), while being asked to ‘Take a seat’ sounds usual/natural. The only other noun/verb pair I can think of which we would use here is bath/bathe. Saying ‘Have a bath’ sounds usual/natural, but saying ‘Bathe’ by itself sounds really strange, and it’s not helped by adding ‘please’.
Google Ngrams shows that the things we most often take are a look, a step, a seat, a walk, a chance, a turn, a position, a part, a view and a recess. Not all of those are interchangeable with the relevant imperative verb. There’s also ‘have a’, which is most often followed by lot, chance, look, right, place, tendency, number, mind, copy and bearing, which list raises even more questions. The only word on both lists is look. Historically, ‘have a look’ was more common in British English, but this has now been overtaken by ‘take a look’, which was previously often seen as an Americanism.
Slightly related: when I was at primary school I noticed that teachers often said ‘Sit up (straight)’, ‘Sit down’ and ‘Stand up’, and commented on this to a teacher, who explained it is possible to ‘stand down’ from an important position (which I hadn’t encountered at that point of my schooling). But this phrasal verb doesn’t necessarily involve standing at all. One can stand down while staying seated.