The enrichener circuit is separate from the slow jet circuit (pilot circuit). It adds fuel at the same place but draws it in through an inlet in the float bowl (brass circle with hole in it) and air through a separate air jet. The enrichener plunger has a rubber end so when the "choke" is shut, the hole at the bottom of the plunger well is shut. When it opens, fuel is drawn through and mixed with the pilot mixture for an enriched mixture at start up. If you open the throttle while starting, it messes it up. The fuel passage for the enrichener circuit is pretty narrow and can clog and the whole slow jet circuit also is full of places that easily clog. The fuel we are forced to use these days with the ethanol draws moisture and settles and tends to clog small passageways. Techron is available at auto parts stores on a stand alone basis and is decent for cleaning as is Sea Foam. Both are petroleum based products and will do no harm as they work if poured into your tank. The most obvious fix is to clean the carburetors and replace the slow jets at a minimum. I think in your case, the enrichener circuit on one or more carbs may not be working at all and you are running on just the pilot circuit at start up and if the enricheners are gummed, there is a good change the pilot circuits are gummed.
