For budgeting and transaction classification, I know PNC has this. I assume (maybe wrongly) that most major banks have something similar.
Speculating for account syncing, but there is a zillion different provider integrations they would have to maintain for something like a mortgage, and such an account would change hands many times to companies with different APIs.
Although which Amazon product wasn't specified, the AWS console is a shining example of something that does micro frontends badly.
To your point about respectable customers using the API, according to Amazon the majority of customers use the web interface for most things. I forget the number mentioned at the AWS Summit, but it was significant.
Even I work full time on AWS and do most provisioning via API actions, but still end up in the console tracking down various things since the API is very cumbersome.
The AWS console is a jarring experience due to this and, in my opinion, not very well done. Every service has a different theme and it is generally ugly.
From a business standpoint, I doubt many people are turning away from them just because of this though.
Speculating for account syncing, but there is a zillion different provider integrations they would have to maintain for something like a mortgage, and such an account would change hands many times to companies with different APIs.