Why doesn't LCR meter measure expected reactance?
Well it's plausible but the numbers seem suspicious (I think). Your meter's manual refers to measuring capacitance/resistance in "parallel mode" by default, but that this can be changed to "series mode". This suggests that it is trying to compute the equivalent parallel resistance of the network. The real part of the impedance of a parallel RC network is

\$\frac{R}{1+\omega^2R^2C^2}\$

... and this is frequency dependant. The equivalent parallel resistance in this case is constant with frequency (by definition it is R) and this is what I would expect the meter to display. However in a real capacitor, the equivalent shunt resistance is not formed by a real resistor.

It would be interesting to switch to "series mode" if possible and see what the numbers are then.