This question will definite qualify for an "
Ask two gardeners and get three opinions" award. Pile up the weeds and they will compost ... other variations are also available
I used to make good compost - the sort that composted at high temperature and didn't take very long to mature. But it did take a lot of my time looking after the heap - finding enough new material for it when the bin was not very full and I needed to keep the high temperature running, and then having to turn the heap etc.
Now I just pile everything up and leave it for a couple of years ... once the initial 2 years wait is over I make as much compost this way as I would do in a high-temperature method. I have no idea if my compost is "as good" as before, but it qualifies as "good enough" in my book (although I'd be interested to hear if a hot composting method would be better for the plants - no doubt there would be less surviving weed seeds etc. in a high-temperature heap).
So by my reckoning you have:
A chuck-it-on, untidy, pile.
A wooden slatted box, ideally large enough to be efficient, e.g. 1 M Cube
A Dalek
Another variation-on-a-theme, such as a "Hotbin" (insulated box designed to maintain high temperature composting)
Wooden box, with removable "front", allows easy emptying. Three boxes side by side, of reducing size if you like, allows turning one-to-the-next. Reducing size allows for the fact that at each stage a "complete bin full" will shrink in size. When the "filling box" is full then empty out the "completed" box, for use on the garden, and move "middle to completed" and "filling box to middle" and then the "filling box" is ready for more material
Or have two or three wooden slatted boxes where you will fill one of them and when that is full leave it, in situ, to compost and start filling one of the other boxes, in rotation. Ideally "turn" the compost to aerate it, or use a plunger to aerate it in situ (a long rod with barbs on it, push into the compost and then withdraw, the barbs will pull on the compost forcing some gaps into the pile to allow air to circulate)
A benefit of this type is that it can be any size, suitable for your size of garden / plot. However, 1 M Cube is the ideal minimum size as it is big enough to keep the heat in and operate in hot-composting mode.
A Dalek keeps all the messy stuff neat ant tidy. I've never know anyone make good & friable enough compost to be able to shovel it out through the slide-up door at the bottom, i.e. a continuous process. IME it is better to fill, and then when full to lift off the Dalek leaving a neat cone shaped pile of composting material and place the Dalek alongside to start the new pile (or have two Daleks). Supposedly Daleks which sit in the sun get nice and hot and that helps composting process, OTOH they don't let much air in (although I've never found that a problem). Dalek bins [and other types] are often offered by Councils at subsidised prices.
Re: lid. I think you need a lid. Daleks have one as standard, but I would make one for a slatted wooden box. I would leave it off in Summer, to allow rain in, and put it on in Winter to stop the heap becoming water-logged. It Carpet will pass by management's requirements then go with that, it is the ideal filter allowing enough water through, whilst keeping a deluge out.