I will throw in my .02 for what it is worth. I am an electronic technician and EE.

The value of the resistor shunt method is that it is something the average person can understand and do, usually without causing much damage, and it gives accurate results.

Yes, most meters can measure DC current through their probes, but it is my experience that almost no one can do this without blowing the fuse on the meter. The meter needs to have its probes connected correctly and then the meter probes need to be put IN SERIES with the circuit, which is different than how voltage is measured. Easy if you understand but most people not "skilled in the art" do not get this detail. So the result is they put the probe in the current measuring hole, then immediately measure a voltage and blow the fuse. Then they wonder why the meter doesn't work. Even an experienced person can forget where the probes are plugged in and do this. I work with a bunch of professional wiremen and half the meter fuses are blown all the time.

A DC clamp-on meter is a wonderful thing. As was said you have to be careful buying one since the vast majority of clamp-on's measure AC current only, including the ones on eBay advertised as "AC DC CURRENT METER". You have to look at the details. A good cheap one is the Mastech MS2108A which you can get for about $40. It will not read small currents as accurately as the resistor method but will measure them well enough. I used one this weekend to track down a short in some trailer wiring. I just sniffed the current along until it stopped. Like magic. Took 5 minutes and I didn't have to touch a wire.

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1998 C230 330,000 miles (currently dead of second failed EIS, yours will fail too, turning you into the dealer's personal human cash machine)
1988 F150 144,000 miles (leaks all the colors of the rainbow)
Previous stars: 1981 Brava 210,000 miles, 1978 128 150,000 miles, 1977 B200 Van 175,000 miles, 1972 Vega (great, if rusty, car), 1972 Celica, 1986.5 Supra