It's because NDSolve is using different method for these 2 problems. For the first problem, Shooting method, which is for nonlinear boundary value problem of ordinaray differential equation, is used; while for the second problem, FiniteElement method, which supports nonlinear problem as of v12, is used. Different solvers of NDSolve are not equally powerful. For the second problem, we can obtain a result by adjusting the option a little. (The result is not perfect though):
sol = First@
NDSolve[{eqns, ic}, {r[R, t], TRR[R, t], γRR}, {R, 0.1, 1}, {t, 0, 1},
InitialSeeding -> {r[R,t] == 3, TRR[R,t] == 3, γRR[R,t] == 2},
Method -> {FiniteElement, MeshOptions -> MaxCellMeasure -> 0.05}]
PlotSol@sol
