Before anyone can answer this question you have to look at the individual vehicle involved and the person driving it.
Not really, no. Fog lamps are essentially, basically, objectively useless for most drivers in most conditions. This is not a matter of opinion or guess or stacked-up assumptions. Driving conditions exist where fog lamps can be potentially useful, but they are very uncommon -- and even where those conditions exist, most drivers cancel out the potential benefit of fog lamps by driving too fast.
Poorly designed and manufactured fog lamps provide poor lighting and a lot of glare.
That is true.
Fog lights, when properly designed, manufactured, and aimed, do NOT blind oncoming drivers.
That generalization can't be supported by reality, even with the wiggle-room you've afforded yourself by saying "blind" rather than "glare".
They have a very low cut off point that keep the light on the ground essentially shining no higher than the lamp assembly height itself.
However, they tend to be small, which means they have very high luminance, which makes them glaring. It is an error to think that a lamp can be glaring only to those whose eyes intersect the actual beam coming from the lamp.
While designed for use in fog the wide flat beam they provide can be useful in the rain
In heavy rain, fog, snow, dust, and smoke, yes...at extremely low vehicle speeds only.
Wet roads are a vexing challenge. Because they are more specular than dry roads, they seem to "eat up" the light from the driver's headlamps. It doesn't actually get eaten, it actually reflects at an angle equal and opposite to the angle of incidence. That means oncoming drivers get that light in the form of reflected glare. Turning on the fog lamps can marginally improve the driver's ability to see the road, but it adds more reflected glare to other drivers. At best, this is a safety zero-sum.
and yes, even in city driving.
No, it just feels like they help in city driving.
True yellow / amber fogs lamps
There is no such thing as a "true yellow/amber fog lamp", that is a persistent myth. Fog lamps are defined by the beam pattern, not the light color. Amber is not an appropriate color for fog lamps. There is some evidence that
selective yellow fog lamps do work better than white ones, but the effect is small relative to that of the beam pattern characteristics.
HID / blue tinted fog lamps do affect other drivers because ot the way that the blue light scatters
No, this is another myth. Blue-light scatter is called Rayleigh Scattering. It's why the sky is blue, but it only occurs when the diffracting particles and droplets are smaller than the light wavelength. That happens up in the sky, but not down at roadway level. Blue light is, however, difficult for our visual system to process.
Fog lights can also make up for poorly designed headlamps.
No, they cannot. That's simply not correct.
Many of today's manufacturers place style ahead of function. While the headlamps may meet the minimum federal requirements they may still be poor in comparison to headlamps designed for functionality rather than style.
Actually, the average performance of headlamps has been steadily increasing for years. Sure, there have been some vehicles with very poor headlamp performance (many Chrysler cars of the 1990s, for example) but in general today's headlamps are at least as good as the best headlamps of years past. That's the objective reality. Subjective perception is often different, because what inspires subjective assessment of a headlamp as "good" or "bad" differs from what makes a headlamp's objective safety performance good or bad.
The use of fog lamps can also be an aid to drivers who are perfectly capable of night driving and who are legally licensed but have some vision impairments.
This is in the realm of possibility, but the number of cases in which it is actually true is very small (vs. just feeling like it's true).
Again....Well designed, well manufactured, and properly aimed fog lamps will not bother oncoming drivers, so no harm/ no foul.
That simply is not correct, no matter how many times you say "again".
And please folks.....Use bulbs of the wattage and design the manufacturer specifies. Few if any of us are properly qualified to make changes to these lights.....
That's good advice.