It is always the device itself that is the best tester of tubes. But for:
Thermal faults - mostly shorts, sometimes going open. A thermal short (or open) will not show up with a VOM when the tube is cold. Further, a rectifier short, for instance, could destroy a power transformer in very short order if not caught immediately, especially if the device is poorly fused (most are). This can be deadly to vintage equipment.
Cascade effects - this tube is a bit weak, and that tube is a bit weak, and yet a third tube is a bit weak. Now, the entire device is 'not quite right' but no singled tube would explain it. And, in the real world, it is this cascade issue that is most common. And, often, why otherwise salvageable devices find their way onto a shelf or into landfill. Sad.
Generally, a thermal fault would have been screened out of the device at some point, one hopes. But not always. And, generally, thermal faults while not immediate are usually quick enough that they would be caught in the initial testing. But not always.
Leads to a tiny rant on how to test a tube. Guys and Gals! Proper tube testing, even on an emissions-only tester, takes _AT LEAST_ ten (10) minutes per tube. First, the set-up and initial screen for shorts and gas. Then the (so-called) "Quality" test. Then, minutes 2 - 10, watch that meter and see what the tube does as it warms up and equalizes. And, at the end of it all, repeat the shorts/gas screening. Only now, do you have useful information.
NOTE 1: Some cheap testers are not capable of holding a heavy output tube for extended periods as the on-board power-supply is inadequate. Be cautious.
Note 2: A tube tester, even the finest Hickok or AVO tests a tube RELATIVE to an established norm or average. A tube may still function perfectly adequately even if all it does is barely lift the meter.
Note 3: Rectifier tubes are the exception to Note 2. They *must* be 'in the zone', or they will be either passing too much AC, or not enough DC, or both.
Note 4: Which leads to being at least aware that the tester in use needs calibration - if possible, or at least pre-testing with known-good tubes to see where the meter falls. Then, compare the others to that.
Tube testers are screening devices. They will indicate dangerous faults (shorts and gas) not possible by any other means other than possible damage to the "home" unit. And they will indicate absolute faults such as an open filament. And for those heavily into Audio, a vanishingly few will allow actual proper matching. After which, it is the "home unit".
A decent emissions-only tester is a handy device for anyone dabbling seriously in tube equipment. IT *may* (not necessarily will) save a bunch of time if used properly. And, a decent emissions tester will not break the bank, either. Leading, again, to:
Note 5: (Heresy Warning!) The step between a decent emissions-only tester and one that will actually convey enough additional information to be useful is considerable in complexity and in cost. A Heath TC3 for 99-44% of general hobby purposes is as good as any but the very top-of-the-line GM testers. That includes the TV-7 military tester and all its clones. The next step up are those capable of matching - or the modern computerized testers that will plot curves and so forth. So, from roughly $80 - $200 for a good, clean TC3 or-equal, to the $900 - $2,500-and-up for a properly calibrated Hickok 539 series, AVO or Card-Matic, don't bother with the in-between.