After the fifth burner light off does it look and act like the rest of them ? Are they all uniform in appearance ?
No you don't want to leave it like that. As a test method it was ok to do that but not for permanent operation.
On the burners there are carryover bridges that carry the flame down the burner rack to ignite all of the burners. All these have to be carrying the flame, even the 2 to 3 second in the first four is too much delay as the light of should be nearly undetectable from one end to the other.
You have already determined that gas will flow through the 5th burner orifice so the carry over must not be getting to it as it would light off if it were. Look closely at how the burners connect to one another and you should find your problem there. It is a matter of getting them lined up correctly and making sure all the passages are clear. Also it does take the right amount of gas pressure to get it to light off correctly, ( you can't be way off on it )
If I gave you instructions on how to clock your meter to find out what the gas pressure approximately is do you think you could do it ? It only involves turning off other gas appliances and timing the meter while your furnace runs for 1 minute while watching one of the dials on it.
Yes , they will check your incoming pressure and that could be the where the problem is but it really sounds more like a regulator problem to me. As if it is taking the regulator a while to catch up Whit the new pressure demand when the furnace fires.
If you start your furnace and let it run while you are looking at the gas meter in your home and time how long it takes for the smallest dial to make one revolution ( usually the 2 foot dial) and let me know I can tell you if you are getting the approximate pressure to the furnace burners, just make sure the dryer and water heater aren't firing when you are timing it. Also need to know the BTU input of the furnace. You can get that info off of the label inside the furnace by the model number.
Was it a two foot dial ? It will say it near the dial .
A two foot dial spinning for 60 seconds indicates that you are using about 120,000 during that time period. So it looks like the furnace manifold pressure is pretty close.
I'm betting its your gas control valve in the furnace, the regulator isn't working properly, that's what all of this tells me , not excluding making sure that the pilot carry-overs are clear. The only test I can't do is to be on the manifold pressure when it is lighting off, this is where I am guessing it is slow to come up at.