OK, if we have compression and we have strong, correctly timed spark, it's likely we have some kind of carburetion problem. You reported that the spark plug was getting wet, but it isn't entirely clear that it was wet with petrol, given the problem you were having with oil at the time.
I've very occasionally had a problem with a line trimmer that wouldn't fire but had nothing wrong with it. Once I'm completely sure the carburetor is OK (and I'm not really convinced yours is OK), my standard solution is to remove the muffler and apply an electric drill to the flywheel nut. Spin the engine up to 700 rpm for half a minute or so. The only times that hasn't cleared the engine's throat and made it run, has been when I've been missing something obvious, like I've been trying to start it with water in the fuel system or there's a problem in the kill switch circuit that didn't show up in the ignition test. In the case of your carburetor I have no confidence in the primer working properly, but by spinning it fast for half a minute it should start without any choke or priming anyway.
This is not a routine way to start a troublesome Briggs. In this case though I think there is something missing from our communication, rather than a mysterious problem. If my wild guess is correct and the problem is in the carburetor, there's a good chance some fuel will find its way through during the long period of cranking. Removing the muffler for this test is just to cope with the possibility that the muffler is blocked, which we haven't disproved so far.
