Santa and his reindeer have to move at fantastic speed to fulfill their yearly duty, this has been solidly established.

Moving at that fantastic speed requires a lot of energy, and all the cookies Santa (and the reindeer) eat at each stop are barely enough to cope with the energy demand.

Long story short: you can't develop diabetes if you have no excess sugar in your blood.

It's very simple. He puts them in his sack. Whenever he leaves a present under the tree, a space is left in his sack. There is plenty of room for cookies. When he gets back to the North Pole he distributes them among the elves as a reward for all their hard work.

As you may have read, Santa is an elf, as mentioned in this popular Christmas poem:

He had a broad face and a little round belly,
That shook when he laughed, like a bowlful of jelly.
He was chubby and plump, a right jolly old elf,
And I laughed when I saw him, in spite of myself

Elves have been known to eat cookies twice their size and gain no weight.

Santa is Quantum

He is able to do this because Santa's magic is based on quantum mechanics.

He visits everyone's house and nobodies house, all at the same time. Additionally, he eats all of the cookies and none of the cookies all at the same time.

That's why it's so important that all the kids are in bed, or he can't visit them. If someone were to observe him or observe his absence, it would collapse the quantum state and he would either have only visited that one house or visited every other house but that one.

Technically Santa does not actually eat the cookies, although he does feed them to his Alchemist Delight 3000 Matter Warping 3D printer (AD3k). The cookies are raw material for the printer which is used to print all the toys Santa delivers. Because the Printer can transform the raw material into whatever molecular structure is required at the Point of Printing (POP) using Just-In-Time Quantum Transmutation (JITQT), most anything would do for raw material. The cookie and milk thing was just a convenient way to make sure something relatively consistent was on hand for processing. Santa used to use coal but it was pretty messy and heavy to haul around. This was also the reason bad children used to get coal in their stockings. It was really the same present they would have gotten had they been good, but Santa used it as a metaphor for the fact that the child had not put the effort into being a better person and so Santa had not put the effort into transforming the coal to a better present.
Interestingly, the processing of the raw materials into presents uses energy created by a room temperature fusion pre-processor on the AD3k and, since the pre-processing generates far more energy than is required by the JITQT, the additional energy is piped back to power the sleigh. This used to generate a great deal of heat and so Santa had to run the feed line down the chimney for insulation against any heat damage. In recent years though, Santa's R&D team came up with a special super cooled cable made of Nb-Ti fibers in an aluminium and copper matrix that is flexible enough so it can be dropped through a window, door or ventilation duct.

A fun fact about Santa is that he is like a bear, except he hibernates the rest of the year except winter. So he has to eat a whole lot of sugar, fruit, or anything that the good boys and girls live to him so he can stay the rest of the year cozy in his north pole house.

Santa is at minimum a being with access to some technology that lets him, travel faster than the speed of light without destroying everything he interacts with, keep track of every living human simultaneously with enough detail to judge weather they've been good or bad, fit through any space he wants, carry a toy for every child on the planet in a sack - while going faster than light speed mind you -, and the means of producing a toy for every child on the planet based on how good they've been and what they've requested.

Santa is very easily a being on par with or greater than Zeus, the Greek god of thunder. Santa can do whatever he wants. For all we know, Santa dilates time to fit his needs and has replacedadded several of his organs to synthesize all the nutrients he needs from milk and cookies. Santa could be taking several years from his side of things to deliver all of these toys. The milk and cookies could be the only food he has to survive. He could be eating a perfectly healthy diet, living at a perfectly healthy weight, for whatever species a Santa is.

tl:dr; Santa might as well be Thanos or Darkseid. Santa could easily be stronger and more cunning than both too. Santa eats what Santa wants.

The cookies Santa eats don't go into his normal stomach, but into a matter converter that provides most of the energy to lift his sleigh full of toys and power the Tardis-like sack that pulls the right toy for the right child across the fourth dimension.

The preference for chimneys as home entry paths was to disguise the plumes of steam and smoke early versions of the process produced.

As I've pointed out elsewhere, Santa is not sequential, he's (massively) parallel. So just as you need to connect every machine in your Beowulf cluster to a power source, every instance of Santa needs a certain number of cookies, and there are a sufficient number of instances to consume all the cookies without ill effects.

*Note also that Santa is rather like a bear, in that those cookies he gets on Christmas Eve have to last him through the whole year.

Santa doesn't need to eat all the cookies. The non-believers just assume that the parents eat all the cookies while the children are in bed. Or the dog. And believers know it must be Santa and his reindeer.

And who's to say they aren't both right? Santa is the master of stagemanship. He uses the fact that perception and reality often blur to make his mystique what it is today. Who knows how many of the cookies Santa eats. 50%? 10%? .00001%? How can you determine the amount if all the cookies are gone and no one can give you an un-biased answer?

Santa knows how to make a name into a brand, and cookies are only the tip of the iceberg.