Most common cause of hard cold start: operator error (LOL). Cold start procedure: pump primer bulb until firm, hi-idle throttle fully open, engage choke, crank till she catches, then disengage the choke and pull back the throttle, giving her just what she needs to stay running. If she tries to die out on ya, bumping the choke usually helps her recover.
3 things the old girl needs to run: compression, spark, fuel.
Your results with priming the carbs is a strong indication of fuel supply issue, but rule out the other two first, they're easier. If you weren't spraying premix into the carbs, be aware that the motor gets its lubrication from the fuel, so giving it gas without oil is like running the motor with no oil.
Do a compression test first, no point spending time and money on a motor with bad compression. Difference between cylinders is more important than raw PSI readings, and all six cylinders need to be within 7 or 10% of each other. If one or two are way under the others, pull the cylinder head for inspection.
You don't say what year motor, but the spark test depends on whether it has the old-school points/condenser/distributor, or electronic ignition, which appeared on the outboard scene circa mid-70's. If it's electronic, set your inline tester to 7/16", otherwise use 1/4", and look for a sharp blue spark during cranking.
Make sure the fuel line is holding pressure and the gas tank is free from water and debris. Fuel should be fresh 87-octane gasoline mixed in a 50:1 ratio with TCW3 certified oil. These old carbed 2-strokes are more sensitive to fuel quality than modern automotive engines.
Check to make sure all the choke plates are fully engaging and staying closed while cranking. You also need full spark advance for a cold start, so make sure the linkage is intact and the trigger moves smoothly all the way to its stop.
If all this checks out, it's carb rebuild time. Only one way to properly clean a carb: Remove, disassemble, soak overnight in carb cleaner, blow out all passages with compressed air (or aerosol carb cleaner), reassemble with new carb kit, reinstall, link & sync, adjust mixture screws (if equipped).
The carb rebuild will work wonders, just take care to get the float heights set correctly, replace all the o-rings and gaskets, and expecially the float valve needle *and* seat. Drill out the core plugs for access to all the tiny passages. Then be sure all the butterfly valves are parallel, closed at idle and perfectly horizontal (not beyond) at WOT, and that the spark advance begins *before* the throttles start to open.
A lot of good advice from Havoc, I wouldn't leave any of it out.
