Is there anyway to buy/book directly instead going through this Amadeus, and this Sabre software you mention?

Not anymore there isn't. All bookings go through one of the travel systems (Amadeus or Sabre). There's no skirting around it unless you fly a private jet and even then, some share passenger manifests with FAA to clear no-fly lists when they file their IFR plans.

So only the wealthy, can avoid sharing their travel info with whoever else is reading this data... What a shame

Nope. It’s the mainframe software airlines use for “computerized check-in” and the ability to list flights from different airlines for different legs of the same journey, just like banks use mainframes designed for checking accounts or companies use ERP systems or governments make budgets. Outside of actually flying and maintaining planes, these systems are deeply integrated to how airlines coordinate between each other, how airports show flight information and so on. We’ll replace them when the last person uses a physical VISA card (probably never…)

Another way of putting it: whatever replaces these systems will likely be forced to do the same thing, so why bother?

To further this, any systems designed to "replace" these systems must also first talk to these systems and provide identical functionality. This was the case with the old BOSS systems we had to integrate with using term screen scraping and pushing text-based booking codes. The same codes used by agents at the ticking counter when you see them type 500 characters just to look you up in their systems. Only now, they aren't interacting with the BOSS system anymore, it's Amadeus or Sabre they are interacting with.

It's a choice by the airline - if they handle their bookings through it, then also direct bookings will go there because they need to have a shared view such as knowing what seats are available/sold. There are various tiny airlines that book directly, but the vast majority of flights aren't with them.

Fly private?

In some cases, private carriers also share passenger manifests to Sabre along with their approved IFR plan. They call it PNR. Passenger Name Registry.

Fly VFR instead of IFR?

Yup. You have to stay under the airspace but this or GPS is one way of hiding yourself from the major GDS systems.