I built something out with Cognito for my current team. I don't like it much, but we're definitely not spending thousands of dollars on it. Have you heard horror stories about its pricing?
From the AWS pricing page (https://aws.amazon.com/cognito/pricing/), you get 50 MAUs with SAML free and then it's $0.015 per user. We haven't had any issues with this - yet.
I do think your solution sounds nicer, but I'm really not sure looking at your pricing that it's cheaper.