Max, if you have been around board design for much time you surely have a tantalum cap explosion story to relate. Back in the early 90's we used them with abondon. All over the cards of the time.
Once in the lab we had a large epoxy coated tantalum let loose and flaming balls of molten material erupted fom the board. The burning material proceeded to wedge between the wall and a mains electrical conduit. The tantalum flamed away and burnt the wall and left a terrible smell with smoke. The burnt wall was not repaired for years and the site was a constant reminder of the dangers of tantalums installed incorrectly.
On the boards - We fixed it with three terminal caps at the time. The two outside terminals were ground and the center the positive leg. No matter how you installed the caps they were correct by assembly.
The series R of a tantalum is beneficial with LRC filter circuits to control the "Q" and prevent ringing with transients. This is much more space efficient than adding an extra resistor. I agree that most "bypass" applications, right next to an IC should be ceramic. For higher current devices, you may also need a resevoir to "refill" the smaller bypass caps at a larger time constant. That is where the tantalums come in handy.
"makes it tedious to track down a problem suspected to be a bad one. "
It's actually a little worse than that. There are no identifying marks on ceramic capacitors, so it is impossible to know if there has been an assembly snafu.
I have seen that first hand, on SSD drives which were supposed to be hot-pluggable. We had some field returns where the regulator(s) were damaged and I was trying to understand and reproduce the failure. The regulators had recently been changed from medium voltage (5-24V Vin) to low voltage (5-7V Vin) for cost and efficiency reasons. I was able to reproduce a voltage surge (2x Vin) during the hot-plug event, so did our pilot manufacturing/test facility once instrumented to look for it. The 2xVin (=10V) took the low voltage regulators out occasionaly whereas the older medium voltage could sustain the transient event fine. The input and output caps were MLCC.
Many hand held and benchtop dmms have a capacitance test function, but testing the caps in circuit can lead to false readings. Using a Fluke 177 meter on a board I have handy, two 106 tantalums read 17uF & 11uF, and 85uF & 9590uF for the two caps and both lead polarities.
When doing lab work and suspecting bad caps, I would always look at the voltage waveforms on a scope first, if they did not look right I would unsolder the caps and check them out of circuit with a meter (or just try replacing it).
Keep in mind that the value measure in circuit is affected by all the devices and parasitic effects connected to the capacitor
Added after 10 minutes: Also there should be no power applied when you attempt to measure. On larger capacitors it is also a good idea to connect a resistor (say 10K) across the terminals to allow them to discharge before you try to measure.
From a non-EE: how do you test capacitors for functionality?
I know more or less which bits of a multitester measure voltage and resistance, but apart from looking for a short by checking for a low resistance between the ends, what tests could one apply? Is there a special kind of meter? (Especially to check for particular values, rather than go/no go.)