The early days of electric instrument design were all basically focused on increasing output/volume/power. When it comes to pickup design, the output of a passive coil is essentially dictated by the strength of the magnet, the number of turns of wire in the coil, and the spatial relationship between the coil, the magnet, and the strings. If you want more output, you can add turns, use a stronger magnet, and/or get the parts all closer together.
However, all of those options have tradeoffs in terms of tone. More turns means more resistance, so you start to lose high frequencies. Stronger magnet and/or closer to the strings can start to cause unpleasant distortion-like tones. And so on. Early pickup designs are all basically juggling acts of finding which of those tradeoffs don't sound terrible. The decisions being made at that point in history created the basic pickups that we are all familiar with, with their expected tones.
Since then, people have refined the designs and introduced new options (like: active buffers/preamps). These newer ideas are usually aimed at avoiding the tradeoffs that we've all been trained to accept as normal. Hence, the pickups produced generally don't sound like what we're used to! Which, in a sense, is the whole point.
Also I think it's arguable that there's some subjectivity to your claim. I agree that active EMGs are generally similar but they do have several different product lines which sound pretty different, and I wouldn't universally call them all bright. Even on the less bright options the active tech is nice since it isolates the pickup tone from your controls - probably a big part of what gives them the easy to identify EMG-ness in the first place - they sound the same regardless of what you wire them up to.
One other thing to keep in mind: it's easier to take a "bright" pickup and cut out the spectrum you don't want, than it is to take a muddy pickup and add back in what isn't there.
