Why don't many modern cameras have built-in flash?

For dealing with low light, flash is just one option. Wider lenses, slower shutter speeds enabled by stabilization (in lens or in body or on a tripod), and noise reduction in higher ISO settings all provide more creative flexibility. Since many built-in flashes have such limited power, even hot shoe flashes provide more options like tilting and swiveling and zooming. Even high speed sync is common.

Historically, most cameras have not had built in flash. As cameras for casual use became more sophisticated in the 80’s and bb90’s, built in flash became more common on mass market designs. The trend away from built in flash is probably just a reversion to the norm.

Also flash is useless for video. Video is increasingly a relevant use. Even for still photography, there’s nothing wrong with continuous light.

Finally, for manufacturers there are not a lot of positives in providing a flash. But negatives include shortening battery life between charges, increased likelihood of mismatch between user expectations and reality, and all the issues that come from additional complexity of design and manufacturing.

There are compact hot-shoe flashes from many manufacturers that fit easily into a small bag or largish pocket. Even ordinary size flashes take less space than a small tripod or large lens.

Integrated flash is still a thing in cheaper consumer cameras, that are mainly built to be used in full auto.

Integrated flash however has some drawbacks:

• Relatively weak, so anything that is not a close upper body shot is probably out of reach.
• As they are mounted very close to the lens, you not only get very flat lighting, you also risk effects like red eyes
• the proximity to the lens also can be a problem, when trying to mount a very fast portrait lens, which might be big enough to cast a shadow onto the subject.
• Small flash size will only give you super harsh light (softness of light is created by having a relatively large light source)
• Uses quite some energy from the shared main batteries
• Cannot be used as bounce flash, as they are faced straight forward. They would not be powerful enough anyway.

Cheap consumer cams however, are on the verge of becoming obsolete due to cell phone cameras. The added computational power in these can do what the flash in cheap cameras never achieved: create usable images in low light situations due to better sensors and automatic image post processing.

So, yes, small internal flashes are an emergency option only. And the images will be worse than that of a current cell phone in many situations. Most modern mirrorless cams can take images in situations where a small flash would not help much, apart from completely destroying any atmosphere the shot would have.

This means that while in consumer cameras you have a scenario of either add some light via tiny flash, which will most probably result in a about correctly exposed subject while the background drowns in darkness or having no usable image at all - the expectations and capabilities of a modern camera is, that the atmosphere of the scene can be preserved via using higher iso and reducing the resulting noise via processing. For example the Nikon D80 has an iso limit of 1600 (3200 with boost). A Sony a7 III has a limit of 51200 (204800 with boost). And I find that the noise is very well controlled until around 6400. After which it becomes more and more visible and has to be taken care of in post production. 204800 paired with a fast lens will produce an image even in situations where my eye is no longer able to see much. However, the image is nothing I would consider usable due to massive noise.

ISO examples of the a7 III (a current prosumer full frame mirrorless at 2020): https://havecamerawilltravel.com/photographer/sony-a7iii-high-iso-sample-images/

If you would like to enhance such a shot with flash, you would need to do things with the flash that is beyond what a built-in flash can do (using it off-camera, bouncing the flash against a wall or using a softbox, using color correction gels, manually controlling the exact amount of flash to balance it with the ambient light).

The electronics in modern cameras (including the sensor but not restricted to it) have become marvelously efficient, allowing for significant reduction in size including the size of the battery. Electronic flash, in contrast, has not changed a lot with regard to the used energy to the achieved illumination. As a result, you still need charge capacitors of significant size and use up a more significant share of the now smaller battery.

Now battery capacity is determined in numbers of photographs taken, and the standard specifies that every second shot (I think) is taken using flash when available. That makes for comparatively unimpressive battery ratings unless you forego a built-in flash.

Also a flash, efficient as it may be, produces some amount of heat both by itself as well as due to the charging and release circuitry. A smaller camera has smaller heat capacity and will heat up faster. Apart from increased image noise, this may mean that some "transparent" plastic parts are in danger of overheating.

Manufacturers have to decide what features to include and exclude. Cameras have finite size, and some features conflict with others. For example, pop-up flash makes weather sealing more difficult, so they wouldn't be included in a model where weather sealing is more important.

Some users prefer not to have built-in flash. They use external flash, and the built-in flash only serves as an annoyance and extra part to break when it pops up.

Manufacturers make models for different target markets. New cameras are still being announced and released with built-in flash. If it is a feature that is important to you, nothing stops you from purchasing models that include it.

As Kai Mattern says built-in flashes have their limitations. For a snapshot, at less than 3 meter people usually do not mind as their strong point is that they are always available and do not require carrying an extra item and extra weight.

For quality photo's the limitations make it unuseable and therefore professionals generally have no use for such a thing. This means basically that consumer grade cameras all have a built-in flashgun and professional grade cameras do not. The flashgun that is good enough to fulfill the requirements of a professional photographer generally cost more than those consumer grade cameras.