Is it beneficial to leave the drying branches so that the plant can salvage the good juices inside them (I'm not entirely sure that this actually happens, it just seems that's how plants work because there's sap), or should they be removed as soon as it is certain that the branch is going to dry out?
This is the correct place to ask this question. The answer is yes, pruning branches that aren't doing anything for the plant saves the plant energy. The plant will 'prune' itself eventually so if we speed up the process, the plant grows faster. Branches that the plant has already abscised will eventually break off in wind.
Plants can 'tell' when a live branch isn't doing its fair share of the work making food for the plant. When the upkeep of a branch takes more energy than that branch is able to create, the plant will start the process of abscission. Shutting off the vascular system to that branch and rerouting water/chemicals (I hate to say nutrients because that insinuates food) to the leaves getting the most sun and making the most food for the plant to store in the roots, grow more roots. In the fall, this same process of abscission causes the leaves to color, then brown and then fall off. The leaves are the most vulnerable part of a tree and so the tree sheds them instead of allowing them to freeze. It lives off its stored food in the roots all winter. No photosynthesis so no more food making so conserves energy requiring little during the dormant season.
Pruning isn't about topiary. Pruning is being able to take off the branches and foliage that; are too shaded, that rub together or that will be rubbing together (allowing disease to enter the plant), branches growing from shallow angles to the branch it grew from (less strength, down the road could cause breakage) and branches that grow into the plant (usually caused by being bound at the nurseries) and reducing air flow through the canopy.
A good pruner knows what the plant will need to do down the road and yes, it reroutes 'resources' SOONER to the canopy for new leaves, new branches. One caveat, never prune more than 1/3 of the plant. This holds for grasses/lawns as well. Little bits of pruning are better than done all at once. Staking trees is rarely necessary and like a cast on a broken arm cause atrophied trunks and support root systems. When the stake comes off the canopy can THEN be too heavy for the wimpy trunk/roots and blow over. Bare root or blown over trees are the only trees I've ever staked.
Always clean your pruners with alcohol before starting on a different plant. Do not use anvil type pruners, they actually crush the vascular system instead of cutting it cleanly. Use by-pass pruners only. Or a tree saw. Use two cuts for large branches, bottom cut first then the top and final cut. Never 'head' trees unless you are trying to ruin the natural tree's form on purpose. Only thin leaving no stumps. The aim is to allow great air flow through the canopy so that the tree won't be blown over and allow some structure/branching to be visible. Bushes on sticks are not aesthetic nor healthy.
And one MORE thing most people don't understand, is that every branch coming off the trunk will forever and always be at that level, in fact as the diameter grows larger the bottom of the branch will be lower than it originally was as newbie twig. Only the apical or terminal buds cause the tree to get taller.
Sorry for TMI, but when I am out driving around I just get horrified with most pruning, especially by the maintenance crews!! I am on a 'mission'! Grins.