We've had some progress in this issue with the help of Azure Support, but we've not found out what caused it, starting from the 4th of September. We've checked our own data exensively and nothing has changed from our end. For us the only possible conclusion is that something has changed with the consumption and flex consumption plans.
The issue for us is SNAT Port Exhaustion. We've seen improvements after making sure that we're using .NETs IHttpClientFactory everywhere, but we were still having some issues until we migrated from flex consumption to an app service plan. We believe that this is load-related and not something we can change without impacting our customers.
We've reached the conclusion that this change is permanent and that the consumption and flex consumption in its standard form does not fit our use case any longer.
Our next step is to explore the option of using a NAT Gateway.