The voltages are normal. The ballast is just an inductor in series with the lamp. The "55V" of the lamp is not what the power source of the lamp should exhibit, but the voltage that the lamp imposes into the circuit, regardless what the current feed is. Because of that, you need some means to control the current fed into trhe lamp and that is, what the ballast is doing: It imposes a point in the circuit where the current gets restricted to the value the lamp is designed for (1.5A for a 70W lamp), letting the lamp "dictate" its own voltage.
In reality the arc voltage is dependent on the exact arctube temperature, so responds to the real power fed to the lamp, but only after a delay, so the ballast is required anyway.
If the lamp is not there or is not ignited, there is no current passing through the inductor, so no voltage drop across it, so you see the full mains voltage across the lamp socket.
With HPS it is a bit more complicated that the lamp needs some higher voltage in order to breakdown the gasses in the arctube, about 1kV or so, until an arc gets formed in the tube. That is the task of the ignitor, in conjunction with the ballast coil. The ignitor senses there is 100V+ across the lamp, so generates 100V pulses between the "ballast tap" and the lamp end of the coil. The ballast coil is then acting as a step up pulse transformer, boosting the 100V pulses from the ignitor to the 1.5kV or so available at the lamp socket. Once the lamp ignites, the voltage across the lamp gets clamped to the 55V (well it is way lower when the lamp is cold, but rises to the 50..60V as the lamp warms up), so des not reach the 100V threshold of the ignitor, so the ignitor stops generating the pulses and becomes passive, so lets the arc burn in the lamp
Now what could be wrong with your fixture:
1) First look for obvious signs of severe damage, indicating the ballast is gone, like charred insulation on the ballast, awful burning smell or so.
2) It could be a bad lamp (very frequent are broken welds after the lamp is shaken during transport when even partially worn out). If you happen to have a known good lamp, just try it.
3) The ballast winding wire could be broken. But your test has shown the mains is passing, so it seems the ballast is conductive, so this issue is unlikely in your case.
4) The winding may have an internal short circuit, so beside being unable to limit the current, it is unable to step up the ignition pulses. A test for this would be to insert about 100..150W incandescent into the socket. If it glows significantly less than when directly on mains, there is a good chance the choke is not shorted. If the lamp lights nearly full brightness, there is an internal short circuit in the coil. It is better to disconnect the ignitor (disconnecting its Neutral wire is enough) for this test, to prevent igniting an arc within the incandescent.
5) The tap connection going to the ignitor may be broken. To test it, just take a voltage reading across the whole ballast coil (LineIn vs LampOut) and compare to the voltage between the IgnitorTap vs LampOut. The later shoudl be about 5..10%.
6. Check the ignitor, most often the internal resistor/HF choke break open and/or the capacitor loses its capacitance. Connecting a "capacitance range" of a common multimeter between neutral-Tap or Neutral-Lamp (only one of them, you need to try) should give a reading in the 50..500nF ballpark range (broken resistor or RF choke will yield <100pF reading, degraded capacitor <<100 nF reading), but none should be DC conductive (resistance reading >1 MOhm; about 1..10 kOhm means the SIDAC is shorted). Either of these, the ignitor is bad.
Dunno how accessible are its components (some were easy to disassemble), it could be fixed, there are just 4 components: A capacitor (usually in the 100..470 nF ballpark, rated at 250V at least), a few kOhm resistor (in the 1..10kOhm ballpark, rated at 2..5W), a RF choke (some mH, a spool of a thin wire with either open ferrite core or with no core at all; these tend to be extremely sensitive to breaks due to corrosion, because of the very thin wire used; the breakage could sometimes be located and fixed, but sometimes the coil needs to be replaced), then a sidac (a diode-like looking semiconductor device, acting as a voltage triggered switch; it could be tested separately by connecting in series with an 20..60W incandescent, it should act like a dimmer yielding slightly dimmed operation but without any flicker). The sidac very rarely fails on itself, but a humidity may corrode out its leads - that would be quite visible even without any electrical test.
Generally broken ignitor uses to be fixed by just replacing the ignitor (any ignitor designed for a "55V" HPS will work here, the exact wattage does not have to match).
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