[Richard777 (OP)]

A Field Equation (acceleration) may be derived from an equation of stress.

If conditions apply, the “stress field” may be reduced to give the Einstein Field Equation (EFE).

The Schwarzschild metric may be simply obtained from the stress field.

Christoffel symbols may not be required.

[jeffreyH]

I think a bit of history wouldn't go amiss here.
https://en.m.wikipedia.org/wiki/Elwin_Bruno_Christoffel
The Levi-Civita Absolute Differential Calculus is also relevant.
http://www-history.mcs.st-and.ac.uk/Extras/Levi-Civita_Calculus.html
I personally own a reprint of the translation. It is worth reading.