Hondo6

Finding the leak in the old tube is usually easy - just remove the old tube and inflate it somewhat with your frame pump, then look/feel to see where it's leaking. I've always been able to find such a puncture without having to immerse the tire. I also mount tires with the label diametrically opposite to the valve hole. That plus the location of the puncture tells me where to focus the search for embedded debris in the tire. But I generally check the entire inside of the tire, just to be sure.

I carry a spare tube in a small seat bag when I ride. After checking the interior of the tire with a finger (very carefully) to find whatever punctured the tube if it's still embedded in the tire, I then remove anything embedded in the tire and put in the new tube.

Often the location of embedded debris is visually obvious. But if it's not, careful and gentle rubbing with a finger will usually find whatever it was that caused the puncture if it's still embedded. (It can also provide a small blood sacrifice to the tire "gods" if you're careless or work too fast because you're impatient. )

I then carry the dead tube home and make a "patch or pitch" decision. I get puncture flats so infrequently these days it's almost always "pitch".