Let the pump prime and see if it helps. If so you are headed along the correct diagnostic path.
Sure you should do normal maintenance items, but I always had to consider the customers comment that they did not want a hit or miss replace a bunch of parts type of repair, and in most cases they had a legitimate concern.
Not a Honda specialist, but my VX fuel system looks very similar to the Nissan systems of the same vintage.
You are describing symptoms of a bad fuel pump check valve, which was a 10dollar item on the old Nissans that used an external fuel pump. I am not even sure if my VX has an external or internal pump, or if it has a separately replaceable check valve.
Another way to test, is to install a fuel pressure guage and us clamps to isolate the different sections of the fuel system and test the different sections for leakdown.
If the diaphragm in the pressure regulator was bad it would flood the manifold with fuel and the engine would be flooded when it started cold, and not run good at all any other time.
I have seen some badly worn plugs that had no effect on cold starting or normal engine operation, some old 300 ZXs with 100 k on the original plugs with gaps twice the specs that still ran fine.
Not trying to discourage normal maintenance items in any way, just trying to help solve the problem with minimal cost and effort.
If I knew the fuel pressure was OK cold, then I would get into a more diagnostic mode of investigation, compression test, and plugs, timing, smoke test for vacuum leaks, etc.
Even a nasty air filter has little effect on my cars performance because I don't come anywhere near close to 50% of the designed air flow of the induction system in any normal operation. I am not syaing don't replace you air filter, just that there is practically no chance that an air filter is the cause of the symptons listed by the original post.
Real mechanics who care about their customers earn the respect of their patrons by spending their money wisely, using their experience to pinpoint problems with minimal expense.