I built an MVP on a 5 dollar VPS. But I had to recently upgrade to a 12 dollar VPS as some of my analytics process were being stopped because of low memory. Aside from writing better code, I found out that using linux utils/app via system calls is the most effective solution as nothing I can write can ever be even remotely as performant as something like grep.
I am following a similiar path to yours but my stack is a bit complicated.
Python + SQLite on VPS
Vue/Nuxt + Auth on Firebase
The process is extremely hacky and experimental, so it might change. I am not planning to use a proper backend as the Python scripts communicates to the Firebase via its SDK. That is working for me at this moment and eliminates writing boilerplate API code.