What does this question even mean? There isn't a 1:1 ratio of nursing aids to nursing home residents.
>But millennials are having even fewer kids.
Thanks to the world the Baby Boomers helped create. Real wages have flatlined, careers being sold as more meaningful than families, and horrible maternity and paternity laws. Meanwhile, the wealth that a responsible couple generates is subsidizing the unemployed person who is on their fourth child. We basically live in a society that has inverted natural selection. That includes flooding the first world with people from the third world and pretending that we're all interchangeable and that the social conflicts created by multiculturalism aren't real.
Researchers are also working on higher-level cognition tasks. We do not know for certain if there will be breakthroughs but there are many diverse tasks that seem within reach in the next 10-20 years.
What are human capabilities that you believe AI/Robots in 2045 will still likely be lacking?
[1] Dense-Captioning Events in Videos
"...We introduce the task of dense-captioning events, which involves both detecting and describing events in a video. We propose a new model that is able to identify all events in a single pass of the video while simultaneously describing the detected events with natural language.... " https://arxiv.org/abs/1705.00754
In 2045 we may very well have highly capable robots, but IMO there's not a chance they will be widespread to the point originally posited: that they will take care of (most of) the old, and produce (most of the) commodities.
One consistent theme when it comes to futurism - whether it's the Jules Verne vintage or the Popular Mechanics vintage is the consistent under-estimation of how much it costs to change the physical world.
Software will continue to be the largest source of change between now and 2045, because software is cheap and generally does not seek major edits to the physical world. Robots and other physical manifestations of futurism are very different.
The futurists of the past predicted domed cities, flying cars, orbital space resorts, mile-high towers connected by floating trains, robotic servants, and the such - none of it has come to pass, and the present world is dramatically less futuristic-looking than originally predicted, because major edits to the physical world is crazy expensive.
Software has been largely an end-run around this expense, by being (nearly) free to deploy to the masses. Deploying billions of household robots, or something similar, is far less likely to actually happen.
IMO a more realistic way to predict the future of 2045 is to imagine how much technological advancement you can cram into a world that still almost entirely looks like the present day. The physical world changes at a comparatively glacial pace.
If we look at you examples of failed predictions: > domed cities, flying cars, orbital space resorts, mile-high towers connected by floating trains, robotic servants, and the such we notice that the real reason none of these exist is because they are all essentially impossible to achieve with our current technology.
However, robotic servants are partially already here. We can verbally ask a device to play a certain song or order us some loo roll, and it works. We have autonomous cars that can, reasonably safely, drive us wherever we want to go. Elon Musk's new factory is, reportedly, almost entirely automated. And as most of this progress is recent, it seems likely that it will continue to advance for a while.
I will agree that there is significant friction against changes in society, which slow down adoption of new tech, but consider that robots in the context we are discussing are not a consumer good, they are a replacement for employees. So perhaps the most relevant question that will drive or stall adoption is: Is a robot more cost-effective than a human.
I also think watching the extent to which self-driving cars are adopted by delivery/freight/taxi companies, and the fallout from that, will be a fascinating (maybe terrifying) glimpse into that future.
The problem is that extrapolation is fundamentally hard.
These "basic" computer vision tasks seem so trivial compared to the actual AI that the original AI researchers famously assigned solving them to a grad student over summer.
[1] In the past, people had children so someone would be around to take care of them in retirement. Financialization of that process decouples the specific children from the specific retirees, but does not otherwise change the basic dynamic that there has to be a working generation to provide for the retired generation. You can't eat money in your 401k.