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Actually, your proposed metric (with the change of signature) will be a solution to the EFE's in vacuum. In the typical derivation of the Schwarzschild metric you end up with a differential equation of the form
$$
\partial_r ( r f(r)) = 1,
$$
where ##f(r)## is the coefficient of ##- dt^2## in the line element. The solution to this is obviously
$$
r f(r) = r + C,
$$
where ##C## is an integration constant. This tells you nothing about the sign of ##C## and generally you have
$$
f(r) = 1 + \frac{C}{r}.
$$
Asking for the correct behaviour in the weak field limit then identifies ##C = - 2MG##, but there is nothing preventing you from putting ##C## positive apart from the fact that we have not observed anything that actually behaves like that (essentially the weak field limit becomes a limit where a point "mass" repels rather than attracts). Celestial objects would move on approximate hyperbolae rather than approximate ellipses, etc. The case ##C > 0## is therefore just not physically interesting.
$$
\partial_r ( r f(r)) = 1,
$$
where ##f(r)## is the coefficient of ##- dt^2## in the line element. The solution to this is obviously
$$
r f(r) = r + C,
$$
where ##C## is an integration constant. This tells you nothing about the sign of ##C## and generally you have
$$
f(r) = 1 + \frac{C}{r}.
$$
Asking for the correct behaviour in the weak field limit then identifies ##C = - 2MG##, but there is nothing preventing you from putting ##C## positive apart from the fact that we have not observed anything that actually behaves like that (essentially the weak field limit becomes a limit where a point "mass" repels rather than attracts). Celestial objects would move on approximate hyperbolae rather than approximate ellipses, etc. The case ##C > 0## is therefore just not physically interesting.