I'll tell you about growing out a buzz cut. You will have about a million layers. It will cost you at least 1 year of your life at neck length to gather about half of all those layers. Then 6 months later, if you're very lucky, you will have to maintain at between shoulder and APL for a while to get just 3 layers out. On your journey you will suddenly want to chop about 3 inches off to get more of the layers out. You will still have layers....It will take you at least 3 years to get near BSL. With layers. That's the layer story.
Now, to the damage story. No matter how careful you are, YOU WILL GET DAMAGE at the awkward lengths. Hair too short to put up, but brushes against collar, getting caught everywhere etc. With your thickness, you'll probably get a raging headache when you can just get your hair in a pony. I know I did. Your canopy is the last layer to grow out. It will be FRIED!
Yes, you will save a fortune in products. But, You will also freeze in the winter and fry in the summer. One month after you buzz, you'll sit down and cry. You will hate the sight of yourself. You will be very lucky if you are at BSL 5 years after a buzz no matter how fast your hair grows.
It's been 8 years since I buzzed and I'm still fighting a fried canopy. I'm only at BSL+.
Of course, it's your head and you can do what you want.
If you really want a change, have you considered trimming to hip length?
If you are on a minimalist kick, long hair is actually easier and cheaper to maintain.