O.K. Here is more than enough info, such that you should now be able to trace your problem on that unit,
It possibly ? being a Luxo or Magna Lux unit with a Circline lamp surrounding a geee-iant magnifier lens mounted in the center.
Take an ohmmeter and place one lead on the [BLACK], HOT, AC line at the plug and then use the other ohmmeter lead to make your first check past the switch and flip to see if the switch is working O.K.
Then test on past the ballast, which you already confirm as having continuity and about the proper resistance already.
The next point would be to either pull the tubes white 4 pin side connector and measure the tubes 4 connections to see if you have two sets of continuity betwen the two pre-heat filaments that the diagram shows. OR, you could check at the socketry, for continuity thru that [A] pre heat filament ,if easy enough to get access to, or else use needle tips to pierce the wire insulation, or at least, those finer points may reach up into the socketry better.
Next side track over to the other prong of the AC plug and check for continuity between that point up to the [B] pre-heat filament in the tube.
If you have continuity thru those two loops, its logical that the unit might have fault with the starter unit or the wrong wattage of starter is being used.
One can eliminate the starter question by getting it where you can unplug the starter and place a jumper clip [BLUE] across one of its sockets terminals, plug in the lamp unit and enact a short connection between the starter socket for about a 5 count, where you should see a weak orange glow inside the lamp and then when the shorting connection is broken the unit should then fire-ionize between the the lamps internal electrodes- and light up.
If not then I would suspect the lamp, particularly if it is 22 years old and has darkened / black / gray ion burns on the white phospor areas near its pre heat filament electrodes.
If you are using the "shorting across of the starter" technique, here is what is happening;
The power comes in and hits the ballast which is performing two tasks, initially it is presenting such a restrictive AC impedance to the incoming 60~ AC power that it is functioning like a heavy power resistor in the manner of precisely current limiting the amount of power that will be able to be consumed in that closed lamp loop circuitry path.
If you will then track the red arrow path, starting at the HOT wire,you will see that it initially gets thru the switch allright and then gets as far as the [A] filament where is sees that there is no place to go across that wi i i ide [A] --[B] chasm there, so it flows on thru the [A] filament down thru the [GOLD] arrow path and ends up at the starter, inside that starter we find it to be akin to a glorified NE-2 lamp with an addition of argon gas and a trace strontium 90 ionization enhancer, along with two hefty internal electrodes, and in between the two is a bimetallic strip that is spot welded to one and almost makes contact to the other electrode.
All we need to do now is look across to the opposite electrode and see that it has continuity to the other side of the AC line. . . . .so we have a potential working neon lamp circuit now, since the voltage IS above ~65 VAC, the gas ionizes and fires and we have a working lamp, but being hopped up quite a bit from your normal neon, so much in fact, that the heat of that ionized gas path additionally rapidly heats up the bimetallic strip and it stretches over and shorts across the starters electrodes.
Well, that then completes a path with AC power thru the ballast and the two pre heater filaments of the tubes with the then limited power that the ballast is letting thru, But wait, lets look back to that starter internals where you had the short betwen the electrodes by the heated and then flexed bi metallic stip, well, with those electrodes being shorted across, there is no longer any heat being produced by the ionized gas path across the electrodes.
Sooooooo the now cooled down strip comes un-flexed and opens its shorting action. That action produces yet another effect , the opening of the complete circuit loop thru the ballast and the lamps two filaments. . . .AND the most important effect is that abrupt disruption of power caused one of the healthy AC fields being developed within the ballast to collapse and develop a multi hundred volt back EMF spike.
Since it is swinging thru 60 positive and negative nodes per second, the odds are good that the timing is such that the disruption will be at a peak voltage level or well enough on up the voltage curve, to produce one HEALTHY spike.
Now if you remember when that lower AC voltage ran across that A-B filament path , it initially ignored that cold tube,and went down the [GOLD] arrow path, but now, with that initial pre-warm up of the filaments PLUS there also being 120 VAC. . . ACROSS. . . the tube electodes to boot, all it now takes is that high voltage spike to initially ionize a path across [A]-[B] electrodes and THEN the current limited 120VAC supply will travel across that just created plasma path.
With that happening, you can then say. . .And let there be light !
Meanwhile . .back in the jungle. . .errrr. . .down at the starter unit , with that new conductive circuit path having been made across the fluorescent tubes [A]-[B]electrodes and it also shunting the starter, that starter no longer has the voltage presence to ionize its gas, so it just remains dormant and out of circuit until power down and cool down of the main lamp.
Remember when I mentioned the ballast having to be on or almost to a peak + or - node when the starter opened up, sometimes that might occur at zero crossover, and in which case, the starter cycles and tries again. . .(and again.)
Manual Start:
If you are using a momentary push-button switch in place of the starter , you are needing it on just enough time to preheat the filament and then its opening action is creating the ionizing high voltage back EMF spike required to fire between the electrodes.
BUT seems like I would say that the FAILURE for a starter unit to initially start , is about the INVERSE odds that you sometimes experience on the turning on of an incandescent lamp, whereupon you flip it on JUST at the peak of an AC + or - voltage node and the lamp pops just like a flash bulb.
73's de Edd