The fastest way to remove bones from a man

Try to get your hands on an Ochre Jelly. Because of their splitting ability, you can cut one up until it hits Small size. That should be managable enough to put it in a large bag (they can squeeze through 1 inch openings, so getting them in should be doable if you pour it through a funnel.) Make sure to have some very small airholes, as Oozes do need to breathe.

When you obtain some fresh specimen, just pour out the contents of the bag, let your little friend eat, and then scoop it back into the bag. You'll need some acid resistant gloves (or just a lot of spares) and proficiency in Diplomacy to convince your allies this is a good idea, but an Ochre Jelly should be able to strip the flesh off a dead body in a few minutes max.

(A living one, too.)

My DM is open-minded, but for the sake of improv, you have to provide some kind of hook. Your answer may assume that you are working with a permissive DM, but RAW answers are also welcome. Spells are preferred, since they have rules-affecting text, but tools are welcome so long as it can be reasonably argued to be quick.

Going by this, you could ask your GM to allow you to research a spell from Pathfinder that does just what you want. It is called decompose corpse and turns a corpse into a skeleton by destroying the flesh.

The classic non-magical method of completely cleaning the meat off a skeleton (human or otherwise) is to first boil the carcass until the meat comes off the bones, then leave the bones and remaining scraps of tissue in a tight container with a species of beetle that will efficiently clean off any remaining flesh. The problem with this is that the whole process takes a couple weeks.

If "mostly clean" is okay, a common butchering operation is "deboning" -- usually done with a "boning knife." An experienced butcher can debone a pig weighing as much as an average human in a few hours (though doing this every day was what led to the discovery of carpal tunnel syndrome). The bones produced will still have a few pounds of connective tissue and a thin film of actual muscle after deboning, but will be far too clean to be reasonably considered to produce a zombie.

To get faster than this, you'll need magic.

You're a necromancer! Let your minions do the dirty work.

At first, this may require you to raise a corpse or two as zombies, or find a preexisting skeleton through some grave robbing. They won't be the most dexterous, but they can easily do the job of stripping off most of the meat. The more minions you have, up to about 4-5, the faster this will be.

Once you've cleaned one and raised it as a skeleton, have it begin cleaning the zombie(s) as well. This may require another casting of Animate Dead to convert the now non-zombies into skeletons, but with a lenient DM, he may allow a conversion automatically.

Once you have a team of dexterous skeletons, you can have them quickly clean a corpse, then take the bones and "collapse" back to whatever storage medium you use for them. You can then cast Animate Dead at your leisure.

Note this does take a small amount of set up and tolerating one or more zombies for a bit, but it is one of the more practical methods available to a Necromancer, other than the lovely answer about Fabricate, which you notably won't have until level seven at earliest.

Obtaining quippers would most likely be one of the fastest way.

In the DMG 5e p336

A quipper is a carnivorous fish with sharp teeth.

They are described as the equivalent of real life piranhas. This way you can have the quippers eat the flesh off of your corpses.

This might or might not fit your campaign depending upon how often you'll be near bodies of water. (Quippers can only breathe underwater).

However, if you have a bag of holding you can potentially hold some water with quippers inside.

Since you are specifically allowing a for a lenient DM, some possibilities spring to mind, all requiring DM approval.

Acid Splash is a cantrip that you can cast over and over again at no resource cost. Whether the acid produced is strong enough to dissolve flesh, yet weak enough to leave the bones is something to discuss with your DM.

Alternatively, consult an alchemist for a type of agent that will work, apparently a strong base instead of an acid. that will work. You'll likely need a prep tank though, so it's not a good on-the-fly solution.

The dividing line between zombie and skeleton is really the amount of flesh. Bones with a few tendons would be hard to justify as a zombie. If you practice, you might be able to quickly carve enough flesh away to qualify. If your DM rules leniently, this might even be a task that can be given to an Unseen Servant.

Prestidigitation allows you to "instantaneously clean or soil an object no larger than 1 cubic foot." I'm pretty sure that this is meant to do laundry, but if your DM is a permissive as you say, you could use this to clean the bones of flesh.

The insect world has much to offer you here.

Here is a fun idea (from a not at all rule minding occasional DM):

1. Make a familiar out of an oversized scarab beetle ala the Brendan Frasier mummy movies.

2. Or make a familiar out of a swarm of such flesh devouring beetles.

Let the Minions do it. Let bugs and other critters do it. Blend the two and keep an army of small, fast, flesh devouring buddies that can also act as tiny scouts. I'm trying to mix Terry Pratchett's description of a beehive as a single organism into this mentally (hundreds of bees, one swarm)

Or have an Imp familiar that can conjure large amounts of ravenous maggots. Make it the really dark side of medicinal use of maggots for cleaning burns and such.

If you have a relaxed DM, you can make the case that this could work in service to the story and story trumps all else.