The ideal continuity tester puts out a low enough voltage on the probes that it doesn't turn on semiconductor junctions, and also has a low enough resistance threshold that it only registers direct connections, not lowish value resistors, most DMMs fail here.
They do, but in many cases this is easily remedied by shunting them with a resistor just above the threshold. So I have a DMM with a 1mA test current, an open voltage higher than I'd like and selectable 1/10/100/1000 continuity thresholds. I shunt it with 10R, which gives me an OC voltage of only 10mV and a threshold error of about 10%, which still works fine for continuity purposes--1.1 vs 1 ohm doesn't matter too much. I would think that a purpose-made continuity tester might simply use this shunt technique to minimize OC voltage, which is a real issue when testing populated boards for short.
I agree, shunting methods work for some specific and controlled circumstances, for instance dry circuit testing where you want a very low test voltage to evaluate contact oxidation etc on relay contacts.
The problem you have with using this method with a DMM for continuity testing is:
1). Any shunt resistor below, say, 150R is going to continuously sound the DMM continuity bleeper - annoying, and the OP wants audible continuity testing.
2). Energised circuit protection. A low value shunt resistor is just crying out to be fried if you accidentally apply it to an energised circuit, probably taking out any components that happen to get in the way at the same time. Fuse or PTC protection are both tedious and potentially expensive or slow acting. They both add series resistance to your probes, which could of course be compensated, but, in the case of a PTC thermistor have long recovery time.
As I said, dry circuit testers definitely have their place for contact evaluation, but can be a pain for general continuity testing, unless specifically designed with sensitive thresholds and protection.
The EE circuit that I posted uses less than 300mV for convenience but this could be made much lower with a smaller energising current on the bridge and higher precision opamp. The bridge approach also offers non destructive energised circuit protection to at least 30V, easily made much higher if you increase a few resistor wattages.