I have some doubts about the sozu proxy that is under AGPL license. If I use the sozu binary as a proxy for my products (API, WebApp, etc.) without any mods or extensions, should I release my products also under the same license or can I leave them privately licensed?
You can probably use the software.
The AGPL-3.0 is exactly equivalent to the GPL-3.0, except for section 13 which only applies “if you modify the Program”. You have not modified the AGPL-covered program. Thus, you can treat it as effectively GPL-3.0-covered.
The GPL-3.0 license would not require you to publish your code just because you use GPL-3.0 covered programs on your backend.
However, if you were to modify Sozu, or were to incorporate parts of Sozu code into your software, then running this modified Sozu software on your backend would trigger AGPL section 13, and you would have to offer your users the source code for the modified Sozu version (but not the source code for other programs you're using on your backend).
So the AGPL is entirely unproblematic for separate services that you use on your backend without modification. Back when MongoDB was still AGPL-licensed, people used it without blinking an eye. The situation is not materially different for a reverse proxy. But if you're concerned about this, use a different reverse proxy under a more permissive license, for example Nginx (Open Source edition).