I took a different approach but probably not very different in age with my kids. I left an ipad in plain sight and let them do whatever they wanted with it. It had restrictions for in app purchases, but left open otherwise and treated as though it was exactly what an ipad is - useless garbage. I'm exaggerating a little, but in my experience the tablets really did nothing except give them something to watch while eating. I didn't want them to believe this thing was special, or desirable, or a treat to be earned. I wanted them to see it like they see candy in our house. Like it was nothing special. I leave candy out readily available with zero restrictions. Always have. The psychological approach to this is to devalue the appeal of unhealthy obsessions early so they can be left in trust without over use or breakdowns or what. And so far we have been exceptionally successful on that front.
Here it is now some 6 years later and as we believed would come to be, the ipads are not a crutch for the kids. They do like them, but they don't over abuse them. They play well in all other ways and treat the ipads like they treat candy - like they're nothing special. The biggest part of our molding with these devices was to use them with the kids, explain things, and not get all excited about any of it. It would be very difficult to recollect every nuance of our ipad and candy history, but it suffices to say whatever our own goals were, we managed to achieve them and are very happy with our kids abilities to use technology and not obsess over them.
There's a lot of discussion many have with concepts of screen time limitations, circadian rhythms, addiction, etc. I have always considered those discussions to be grasping at straws. People trying to brand themselves as authorities on the subject. Like many things, I take it all with a grain of salt and conduct my own experiments. I am happy with my own results.
This is not to say they are wrong. Just that you don't need to live a stereotype where you adhere to the popular method and just trust "experts" over your own ability to observe and react. After all, this is your child. You know them best. And while there may be a certain degree of axiomatic science behind many claims, they are also applied to one of the most diverse mysteries of history - the human mind as it adapts with the times around it.
That is to say we are moving fast in an age of technology. Screens are inevitable. Responsible use is mandatory. You should teach your kid and play with them in all ways with technology the same way you would not just assume they will learn to ride a bike or drive a car alone. A little discussion, questions, answers, and the joy of mystery in life can clear a path to - if nothing else - a slightly more emotionally balanced approach to the bright and shiny.
I disagree to this trend of "no screen time" as you have stated because I do believe what you said in the last part of your question is true: "By keeping screens from her so stringently, is she more likely to be obsessed with screen time later?"
Yes, I believe that may be the case. But it is not just screen time. It is anything you adamantly reject as an option for their free minds to explore. People tend to fixate on the "off limits" and when they get little doses it might be very hard for them to understand why they can't just have it all the time.
I'll probably be the outcast in this matter of opinion - and it is just that - my opinion which is based on my own history and success in circumventing unwanted attraction to items of common addiction. I am not a medical doctor, not a psychologist. I just observe and adapt like many of us out there and this ramble is what I have come to find in my tenure as a parent. Hopefully it is useful to someone.