Hi Don,
Sorry to hear you're having these problems. I'm also sorry to hear that you are using starting fluid on this fine engine. I physically flinched each time I read "used starting fluid". I absolutely hate the stuff. It really can't be used without causing some damage.
Whereas gasoline and other motor fuels ignite and create what is known as a "flame front" that burns evenly and builds pressure in the combustion chamber, starting fluid, or ether, is much more volatile and is more like an explosion when it ignites. And, as with any explosion, the resultant violence usually removes a bit of metal close to the point of ignition.
Just a few applications of this stuff can ruin an engine and turn it into an "ether baby" making it act much like a drug addict that needs more and more of the stuff to get going. Anyway, that's my speech, now back to your problem.
It does sound as if you still have carburetor problems and that is what I will focus on here. You seem to be convinced that's what it is. But, you might also have an ignition problem causing your hard start. It is always a good idea to use fresh plugs and verify spark when problems like this arise. The fact that it runs after a bit of coaxing does seem to indicate a fuel issue.
I have actually taken a brand new float chamber out of the sealed plastic bag it came in from Honda and found casting sand packed in the passages at the bottom of the bowl! Sad but true. I very carefully go over new parts and clean them prior to installation after that fiasco. Perhaps something similar has happened to you with your new carb.
Do you have any idea what caused or where the "goo" you found came from?
Don't worry too much about the accelerator pump piston. If it depresses and then returns, it is ok. What you need to focus on is the passage in the body of the carb that it feeds. You may need to remove that tiny screw to decipher how that pump piston feeds the carb. but an easy way to tell is, with the float chamber in hand, supply some WD-40 or carb spray to the pump cavity and depress the piston with a small screwdriver. The orifice that the liquid comes out of will match a passage in the main body. Use a can of carb spray and it's little straw and BLAST that passage. You should get a good shot of spray from the jet orifice (if you removed the jet set) and the main and intermediate orifices at the top of the carb throat just behind the throttle plate. I use a pocket screwdriver wedged in the throttle shaft linkage to hold the plate open so all three orifices can be seen. The idle, the intermediate (transition) and the high speed orifices are in a little triangle at the top of the carb throat.
Yes, the idle screw needle valve is under that plastic limiter cap. I usually just snap it off and get busy. If you snap off the limiter, you will usually also snap off the slotted head of the needle valve with it. I use a very sharp dental pick to grab the smooth outer edge of the needle valve screw and rotate it counter clockwise to remove it. It is a pain to get out but once it is, you can file or cut a slot in the head and reuse it. I just use a new needle. THIS IS THE ONLY WAY TO COMPLETELY CLEAN THE CARB! If you don't remove the idle screw and get that passage flushed out, you will always have problems.
To flush and BACKFLUSH all the passages so that they are reliably clean, you need to hold your fingers over passages and plug them as you are spraying others. The enrichment passage that goes to the bystarter needs to be cleaned as well and it's the same deal. All of these passages are interconnected with one another and you need to plug and spray in one combination and in one direction and then plug and spray in a different combination and in the other direction to get them clean.
For example; if you stick the carb spray straw up into the cavity of the carb body where the jet set is inserted and hold your fingers over the passage for the accelerator pump AND the cavity for the idle mixture screw, you should get a good 6 to 10 foot stream of fluid out the back side of the carb throat if you have the throttle plate propped open. If not, then place the tip of the straw over each of the three orifices, one at a time and backflush. It can take several attempts and two or three cans of carb spray with a very dirty carb.
Yes, the passages at the bottom of the float chamber need to be clear too. Make sure a small strand of copper wire can be inserted all the way through each passage from one end to the other. One is just the drain but there is a vacuum break passage and the accelerator feed passage that are both critical.
I predict that you didn't need a new carb so hang on to your old one for a spare.
Now, after I wrote all that garbage, I want you to do one thing first before doing anything else.
Refer to the parts page link below and look at items #2 and 3. Take that thing off and shake all the liquid out of it and wash it out with fresh gasoline. While you have it off, briefly crank the engine and make sure your fuel pump is delivering fuel. Put that fuel chamber back on and see if it doesn't start better. If it does, you're done. If not refer to the verbiage above and good luck.
http://www.boats.net/parts/search/Honda/Outboard Engine/2006/BF20D6 SRTA /FUEL PUMP/parts.html