A public offering is the company saying "here is ownership in exchange for your money". A share buyback is the company saying "here is money in exchange for your ownership."
You're getting downvoted because it doesn't make sense for a company to be able to move in one direction, but not the other.
Right, they can buyback, but no matter how far they go in that direction they won't end up as a private company.
The shares they purchase don't transfer the ownership they imbue to the board of the company -- they are either retired, held as treasury stock or given to other shareholders. The majority shareholders remain majority shareholders of a smaller amount of stock.
To think of it in a very simplified way, if the company bought back all the shares except 10 from some staunch holdout, that guy would own 100% of the shares of Apple, (and each share would have a stratospheric price).
For the board/management to take control of Apple from the shareholders (which is what people mean when they talk about going private, really), they have to personally acquire more shares.
They obviously can't do this with the company's money, because shares bought with that transfer to the company and either dissolve or are transferred. They have to use outside finance to do it.
This is what I mean about you can't buy the company from the shareholders with their own money.
(They could in the simplified world use the money to buy back the stock and grant it all to board members, but in this world there is a tonne of legislation preventing this, and if they did it to take control out of the hands of shareholders they'd be open to action.)
You're getting downvoted because it doesn't make sense for a company to be able to move in one direction, but not the other.