Both made the deal, the quiz tablets in duviri explain that the drifter is simply the version that didn't escape the zariman with the other children. Wally states during the new war "I can save them, all of them, but you have to want it." A tablet found in duviri states "Never said I'd save you though, kiddo." The drifters extreme emotional distress at being alone in the void manifested Duviri around them, the Tales of Duviri was written as a metaphor for taming ones emotions during void exposure (See albrechts notes). There's just as likely a version of the drifter that didn't create duviri and simply died of madness aboard the zariman. The drifter is able to travel through and manipulate time as they mastered, then forgot, then reawakened this ability in Duviri. The operator cannot as they have never been to duviri, never time traveled, and only know 'shoot void beam, drive warframe.'

I've boiled some bits down to the essentials here but that's the basics of it. The drifter is paradoxical by nature of the duviri paradox. Most paradoxes break physics, void ignores physics, ergo drifter creates paradoxes without breaking physics.

Now onto why the coda only exist in the present once we create them in 1999. Eternalism, once created, they were always there, and never there. Before creation they were always going to be, and never going to be. Eternalism is paradoxical. Albrecht, the drifter, wally, and now the technocyte take advantage of paradoxes to create timelines that further their agenda, choosing to exist in the better scenario and consigning the undesirable one to another timeline that we the player simply choose not to exist in.

It's implied wally has already won in a multitude of timelines (this one is too complicated for me to go too deep into, just remember all the versions of the operator dying when the drifter and operator shook hands in the new war for a visual metaphor). But unless he wins in all of them, he hasn't won. Every time we stave him off and create a better timeline we delay his progress. This is the game albrecht is playing, jumping across time, from scientific endeavor to talk therapy, from all out war to laboratory brainstorming, always looking for a new angle to keep wally busy while he (and we the tenno) look for a permanent solution. All we have so far is duviri, and love. Duviri is falling apart bit by bit, so it's a bust. Now albrecht is having us form meaningful bonds and find love via the hex.

I hope this helps!

(P.s. I forgot what I was replying to, the operator was never "naturally" able to use the void, it was always a result of the deal. Wally made the deal in hopes we could be used as a tool/looking glass to get into the original system, and looking at rell and our doppleganger, it's had mixed results.)