@David Gonzalez D Thank you for reaching out to Microsoft Q&A. Based on my understanding, you are looking to establish +20,000 concurrent connections to backend API(s) (here: App Service) and observed some issues with API Management.
As per APIM limits docs, for premium tier the limit is 2,048 per unit and you can add up to maximum of 12 per region. Can you let me know how many units you have tested with? With 10 scale units, you can handle 20,000 concurrent connections. Please note, Connections are pooled and reused unless explicitly closed by the back end.
Also check Pricing page for more info regarding this and if you have a demand to increase more, our Azure Support can further help you with that. Feel free to let me know if you want me to connect you with them.