In other words:    Does water boil faster with the lid on? Yes. But folks often don't consider food looks cooked 'till it's browned, and will find that takes a really long time to look cooked if they don't take the lid off.  How can these both be true?  
Your question is a little odd: you first ask about two things to do with cooking in water, then you ask about browning, which never happens when cooking in water no matter how hard you try. You need higher temperatures than the 100C you can reach in boiling water.   There's sort of an exception: if food sticks to the bottom of the pot when boiling, or if the liquid is pretty thick (like a thick sauce), it can burn on the bottom. This is pretty easy to do if you're not stirring - the bottom of the pot gets well over 100C without water directly in contact with it - but that's usually more of a way to ruin your food.  So I don't really know what to say about the "how can both be true" question - boiling faster with the lid on has nothing to do with browning. The former is about the speed of adding heat, and the latter is about the actual temperature. They're just two completely different things.  As for the question in the title, the answer is... sort of. High heat and putting the lid on both will let you reach a boil faster, which means a shorter total cooking time since you're spending less time at lower temperatures. And they also both make it easier to make sure the whole pot is boiling. But once you reach that point, no, adding extra heat or putting the lid on won't make things go any faster.  But you might be down in that "not quite all boiling" range more often than you think. This Food Lab article on boiling has a great photo:    Remember, heat is only being added at the bottom. So the water boils, turning into steam, only at the bottom of the pot. That part is definitely at 100C. For the whole pot to also be at 100C, that heat has to be transferred to the rest of the liquid. With a full, rolling boil, that happens: the water and steam are mixed around, coming to the same temperature throughout. With a simmer or low boil, especially in a deeper pot with more food and less water, or with thicker liquids, it's very easy for that not to be the case. If you have a pot of a thick sauce at a low boil, or a pot with more food than water, with some small bubbles coming up, you might see temperatures perhaps down in the 80-90C range at the top. That does mean somewhat slower cooking! Turning up the heat to get a stronger boil will bring that temperature back up, as will putting on a lid to hold in heat. 