I'm a player that tends to use chess.com computer analysis on games I find somewhat interesting just to see which moves it sees as immediately poor, and mostly to check my continuations after opponents make mistakes to see what kind of accuracy I can play with. (All of my computer software is located on another computer that isn't currently connected to the internet for me to download the pgn and do my own analysis there.)

So the question is, when I'm some absurd amount of material up, like N+p and good king position or better, and I trade down from being up a ton of material to a position that I've judged that even -I- can't loose, should I heed the computer calling mistake after mistake, or be happy that even if I'm not efficient, I'm still winning a game?

For example:

Here I noticed that my connected passed pawns in the middle ensured a win if all pieces are traded because I was confident my opponent would allow my g-pawn to freeze his g and h pawns. The game proceeded along those lines, and it ended pretty much exactly that way, though there were clearly ways to win with the pieces, if only I had taken the time to find them.

So, am I right for going for a "dream" position and grabbing the win, or am I right for taking the worst case win but very very wrong by being too lazy/incompetent to calculate winning variations when up the piece? Or, am I just lucky that I made the 2nd to last mistake?

If you have accurately calculated that you win easily even if you sacrifice some of your extra material back then there's nothing wrong with that. Indeed, sometimes this is the easiest way to win.

There's no way to win with the pieces. The pieces are supposed to support the promotion of the pawn(s). Your opponent of course will prevent the advancement of your pawns (and do his own pawn advancement). He will do it using his pieces. So of course you have to remove (exchange) them when they get in your way.

Not focusing in advancing the pawn, but on pieces maneuvering may give you trouble. In situation like this the winner is usually the one who can promote the pawn faster. I have seen enough where the side with more material loses because he underestimates the pawns advancement.

Computer analysis is not always accurate ... and sometimes it can be not only unhelpful, but can even be detrimental to good learning.

It is perfectly sound to deliberately trade off material in order to reach a won endgame... even if the computer analysis says there is better.  A win is a win. 

If I had a choice between trading off to a simple won endgame where mistakes are highly unlikely and a more complex/risky plan of gaining something else that the computer says is evaluated better, then I would take the simple win any day.

always its better to have a clear winning plan with no chance of counterplay. if computer thinks otherwise then computer is stupid. This is not the only situation that computer eval is totally wrong.

probably you have lost from positions before where the computer thought you were totally winning. If you think about it, this is an example of the computer eval being wrong, its not just your fault for messing up somewhere. The only truly 'objective' evaluation is a reasult. Draw in x moves, or win in x moves. Actual evaluations (measured in pawns) is simply a measurement of the likelihood of mistakes...your mistakes should be accounted for in a proper evaluation...but computers dont make the same sorts of mistakes humans make, or as many...and its evaluating for ITSELF, not you.