I just found this solution from a thread in BA.
Route system output (stdout/stderr) of Blender to the app console (put in scripts/startup) · GitHub
I just throw the log.py into the blender\3.0\scripts\modules (installation path is not strict - just anywhere that the sys.path can look can do). And it works.
Why isn’t this superb solution offered in the Blender distro? Everything is great as of now. 
P.S. Though the solution is a workaround, the most proper and standard implementation is to have C code written. But in any case even this workaround if considered safe, it won’t be bad at all.