We consider the capacitance capability in a DMM as an aid rather than an absolute measurement capability, good for "seeing" if a cap is any good, but not characterizing such, altho one can achieve some reasonable results in certain cases.
Same for tweezers, and they cover the inductance as well, but measuring a component in-circuit and expecting an accurate representation is generally wishful thinking. These are great for trouble shooting, and agree they should be in everyones toolkit, or at least the tweezer cable sets that work with the quality LCR meters, altho they aren't as convenient as the standalone tweezers.
One must realize that the DMM measures DUT capacitance by measuring the charge/discharge cycle characteristics with a rectangular excitation and computes the capacitance based upon this, while the quality LCR meter excites the DUT with a sine wave (at various frequencies) and measures the DUT Voltage and Current with high resolution. These measurement V and I results are Vectors (Magnitude & Angle) and utilized to compute the various DUT parameters which include a host of useful parameters and not just capacitance, nor resistance, nor inductance, but various combinations of such.
For general accurate representations of components these must be measured in a standalone controlled electrical environment and/or fixture, and properly calibrated with the environment/fixture effects removed. For the DMM capacitance capability this requires a single Cal by "nulling" out the residual cable capacitance, whereas the LCR Meters and Tweezers require a 2 Cal procedure, Open and Short, some at various frequencies. Also, some quality LCR meters have an additional Ref Cal procedure with a known Precision Reference Device(s), all at various frequencies, and obviously intended for precision work!
The DMM and LCR meters are completely different instruments, which employ completely different techniques and intended for completely different use cases, and Tweezers may help fill the gap between.
Anyway, having a quality DMM, LCR Meter and Tweezers usually covers all bases, as always YMMV!!
Best,