Maybe it's not possible with picking, but having worked at Amazon IXD, I think it's certainly possible with most items (>70%) in the receiving, sorting, and preparation processes at the cross docks.
I'm not sure about Amazon Logistics in the US, but at least in Europe I got the feeling that management just doesn't care about automating processes because humans are still cheaper in most cases.
Amazon Logistics uses some brilliant tactics in Germany to basically get workers for 50% off while maximizing productivity; all subsidized by taxpayers:
In West Germany we have countless unemployed people after the migration wave of 2015/2016, and those who were able to stay and gotten asylum are often "unskilled" workers who have trouble finding a job. These people go to the employment office (Arbeitsamt), and the Arbeitsamt's "Jobcenter" then sends them to Amazon. If people don't comply, the Arbeitsamt cuts their unemployment benefits, which would lead to even worse living conditions for those who are already living in poverty. So, there is a big incentive to take these very bad jobs at Amazon.
The worst thing about this is that in Germany the government pays employers an "integration subsidy" (Eingliederungszuschuss). This is a percentage of wages paid by the state.
It's up to 50% for 12 months for everyone who joined Amazon via the Jobcenter; and for people who were unemployed for >2 years, it's 75%(!) the first year and 50% the second year.
Also, after two years, an employee is entitled to a permanent contract. Everybody wants to have this unlimited contract, but only a small percentage gets it. Therefore, workers with fixed-term contracts try as hard as they can to get this offer at the end of their term.
However, Amazon knows exactly that after the workers get a permanent contract, the productivity decreases. There are internal statistics for this (this is also the reason why only such a small percentage gets this offer in the first place).
But the smart thing now is that each warehouse is a separate legal entity. This means that in reality, workers are terminated after 2 years, but are offered a new contract by the warehouse that is right next to it. There they then have to work extremely hard again for 2 years, hoping for a permanent contract. HAt this state most people give up and leave Amazon.
Anyway, back to automation:
In Europe, Amazon has always just bought extremely expensive instead of investing in research themselves. We then bought overpriced machines like OPEX Sure Sort. The sorting system itself works quite well, but it's just not integrated into the process at all. This means that at the end of the day, people have to fill the machine manually, and the bottom line is that only two workers per machine (which costs a quarter million euros, so you need 4 years for break-even) are eliminated.
The only in-house product we had from Amazon Robotics / Ex-Kiva was a robot for palletizing EPAL pallets. This thing is actually great and really helped to maximize throughput.
I don't think Amazon will ever achieve an all lights off IXD with the current governance.