Most "first timers" don't get these carbs completely clean. They are simply tough little boogers to get all the gunk out. That's why I recommend the carb manual that walks you through it with pictures OR a new carb.
I've had guys here clean them 7-8 or more times and finally get positive results.
My guess is that you either didn't get the passages in the bottom of the float chamber clear or you didn't get the jet-set completely clean. OR...the jet set tube is cracked and you didn't see it.
Those are the areas that cause the most hard/no starts
I didn't re-read all of the above but did you have a 6 to 8 foot stream of cleaner spray come out of the accelerator pump orifice? That's very important.
Did you backflush each drilling at the back of the carb throat?
Did you remove the idle mixture screw and flush the idle passage?
Did you flush the main air and slow air passages from the front mount flange?
See? Those are just SOME of the procedures that need to be done.
You simply can't be expected to know all of this first time out.
As far as the accelerator pump goes, yes it will come with a new carb. But you can also buy the parts for the pump separately. See item 22 and 21 in the link below.
https://www.boats.net/catalog/honda/outboard-by-hp-serial-range/8hp/bf8d3-lha-2003/carburetor-manual
While the pump push rod extends up through the main body it is actually a part of the float chamber. If the pump piston bore is scored (most are) the bore isn't serviceable and a new float chamber is the only solution for a "total" repair along with a new piston and spring.
While the accelerator pump CIRCUIT in the main body is very important to keep clean for reliable starting and running....
....I'm completely confident that the pump itself is NOT causing your start up issue. I have actually run this exact same outboard WITHOUT the piston and spring in the bore. Also, about 100% of the pistons will eventually sieze in the bore and the outboard will still start and run well.
Bottom line? The accelerator pumps on these are almost worthless but the CIRCUIT needs to be clean for the engine to run properly. I admit that I can't fully explain why but I do have plenty of experience with this being the case.
I do wonder if your "choke cable" is working properly. If it isn't, you will have the same problem even with a new carb.