While in Israel and in 2xxV territories in general, capacitors don't affects lamp performance, operating a fluorescent or HID lamp in North America, without a capacitor in the circuit, often results in poor lamp performance.
Why this is the case?

In MOST systems here, capacitor is for power factor correction only, so it is connected to the input power in parallel to everything else : The capacitor getting 230V, and everything else getting 230V. So it works on 230V independently from the capacitor

Exceptions for 230V are Rapid Start fluorescent ballasts (both the "105W" type and SRS type), Leading Fluorescent Switchstart cictuits (rare), and Autoregulator HID ballasts

In MOST systems in the US, capacitor is part of the ballasting impedance, and is in series with the lamp or with one of the coils in the ballast. So changing its value affects the ballasting impedance

Does this explain why some Street lights I see with Capacitors only reach 40-65% of their brightness capacity, and don't reach their full brightness?

Does this explain why some Street lights I see with Capacitors only reach 40-65% of their brightness capacity, and don't reach their full brightness?

Well, what looks like "half brightness" is usually way less - about 20% or even much less. The real half brightness appears as just small difference...

And yes, the worn out capacitors are very often responsible for that.
On top of that is the effect of the lamp state being locked in a low voltage saturated vapor mode: Even when the current is not really that low, it is sufficiently low to not give the lamp enough power, so heat, to evaporate all the dose. The thing is, initially the lamp has rather low voltage drop, so the same current delivers half or even less power than it will do with the fully warmed state. And because of the lower arc loading, the efficacy drops a lot as well, reducing the light output even further.
So I would not see as impossible with 50% arc current from a CWA (so a constant current source) the lamp to not reach even 10% of the rated lumen output.

Notice the one on the left doesn't get any brighter

Is this because of the PFC capacitor failing or losing capacitance? I have seen metal halide lamps cycle on CWA ballasts with bad capacitors. Do these HPS streetlights have CWA ballasts instead of HX?

Generallt rhe capacitor wear translates into capacitance loss, so reduction of the current in the branch with the capacitor.
So if the capacitance is the main ballasting impedance (like in CWA), it has direct impact on the lamp current.
If it is in the cathode preheating path (like the starterless 240V SRS fluorescent circuits), it makes the lamp harder to start but if it does start, it then burns normally. Only if it degrades below a critical point when the preheat (and the bit of voltage boost) are insufficient, it just does not start at all anymore.
And if it is just forming an input power factor correction, the lamp burns normally but the ballast just has worse power factor, so higher input current. With some ballasts (like some multivolt HX autotransformers used for pulse-MH in the US) this may lead to ballast overheating, so reduction of the ballast lifetime.

Is this because of the PFC capacitor failing or losing capacitance? I have seen metal halide lamps cycle on CWA ballasts with bad capacitors. Do these HPS streetlights have CWA ballasts instead of HX?

From what I have seen, most North American high pressure sodium ballasts above 150W S56 are typically CWA.