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

[![enter image description here][1]][1]

"Then, is it possible to avoid using pure `FiniteElement`in `NDSolve`?" Well, for this specific toy problem, if we differentiate the first 2 equations in `t` direction and add proper initial condition, it's possible:

    γθθ[R_, t_] := 4
    tend = 1;
    bleft = 1/10; bright = 1;
    With[{r = r[R, t], TRR = TRR[R, t], γRR = γRR[R, t], γθθ = γθθ[R, t]},
      
      eqns = {r D[r, R] == γRR γθθ R,
              r^4 R γθθ D[TRR, R] == 2 γRR (r^4 - R^4 γθθ^4),
              D[γRR, t] == 0};
      
      bc = {r == 0.1 /. R -> bleft, TRR == 0 /. R -> bright};
      
      ic = γRR == 3 /. t -> 0;];
    
    Block[{γRR}, γRR[R_, t_] := 3; 
      solinitlst = 
       NDSolveValue[{eqns // Most, bc}, {r[R, t], TRR[R, t]}, {R, bleft, bright}]];
    
    icadd = {r[R, 0], TRR[R, 0]} == solinitlst;
    
    sollst = NDSolveValue[{D[eqns // Most, t], eqns // Last, bc, ic, icadd}, {r, 
        TRR, γRR}, {t, 0, tend}, {R, bleft, bright}, 
       Method -> {"MethodOfLines", 
         "SpatialDiscretization" -> {"TensorProductGrid", "MaxPoints" -> 50, 
           "MinPoints" -> 50, "DifferenceOrder" -> 2}}];
    
    Plot[sollst[R, tend] // Through // Evaluate, {R, bleft, bright}]

[![enter image description here][2]][2]

Notice this approach will probably fail in more complicated cases, because the DAE solver `NDSolve` is not that strong. In those cases, advanced technique is needed, [here](https://mathematica.stackexchange.com/a/184285/1871) is an example.

  [1]: https://i.sstatic.net/6felw.png
  [2]: https://i.sstatic.net/PVJo5.png