First of all, thank you for sharing all of this. If more people shared difficult management challenges like this, we would all feel a little bit less like imposters.
The situation you describe isn't uncommon. I think most of us have had a stint in a shop or at least a period where we are asked to perform miracles. We are tasked with this with such confidence that it feels like we aren't up to it. This isn't true. The reason we are asked to perform miracles is because technology is miraculous for most people and the difference between an easy and an impossible project is completely opaque.
Now let me be clear. If you are having a bad time at your job, you should leave. Life is too short to stay with places and people of torture if you can afford not to.
But, if you are willing to give it a shot here's my assessment. The story you're sketching sounds like a turbulent time where the business is growing rapidly and/or has lost Engineering leadership. Engineering leadership is needed to to have realistic projects. If your job is to perform miracles, you will fail every time. What you can do is able to communicate realistic limitations to your stakeholders and negotiate consensus on what can be achieved. You need to create a situation where the business can sacrifice a 20% chance of taking 5 steps forward this quarter for a guarantee of taking a single step. A business that can't see the value in that is dead in the water anyway.
The engineers on the other side, need to have their priorities cleared and set. You need to recognise premature optimisation, hobby projects and resume fluff and put an end to them.
This has nothing to do with Amazon lambda and everything to do with communicating arguments effectively, creating relationships of trust with stakeholders and developers.
I won't lie. This role will cause alot of friction, but it sounds like the business has no one to tap on the breaks. In the end the real miracle would be to turn the boat around and have some predictability in the workplace.