I also prefer to use the throttle (accelerator pump) to prime. Less likely to overprime.
NOoooo.....!
Not on the PA-28-161, at least. A few years ago a renter pilot set the Warrior we were operating on fire during startup. It was only saved buy a nearby pilot who discharged his own extinguisher into the cowling.
After recovering the aircraft we carried out some tests. The pilot had mentioned priming by pumping the throttle and it turned out that four or five pumps of the throttle was enough to have neat fuel running out of the hot air valve under the carb. This assembly is far from sealed and so this fuel ran straight onto the inside of the cowling.
In contrast no reasonable amount of primer use would produce neat fuel running back through the inlet. Apparently the design of the inlet pools the fuel in the head rather than letting it run back.
What started the fire? Backfires are apparently very likely and that's all it would take. If the fuel is already in the engine bay, no amount of 'drawing it back in' etc will help. (edited to say I suppose if the hot air was selected then it might draw the flames in I suppose)
Behind all this in my opinion is the issue of poor starting which seems to be widely accepted and leads to pilots searching for 'innovative' techniques to get going. In my view poor starting is as serious a fault as, for instance, no radio or something like that and needs to be fixed rather than justified as seems to be the habit of some engineers of my former acquaintance.
There's no reason why the engine in a Warrior should not start immediately, hot or cold, and if the aircraft you fly doesn't do that then I suggest complaining until they find out what's really wrong with it rather than accepting the situation and risking, as also happened to the same aircraft, a burned-out starter motor in the Channel Islands, or something much worse!