Why aren't fixed build platform 3D printers popular? I'm trying to build a DIY 3D printer for myself. I've been exploring many different styles of printers and found this type of printer that has a fixed bed that stays fixed in one place and the whole gantry moves which includes all axes.

Why is this so rare?
Are there flaws in this design?
Will print quality be affected by using this approach?

Check out this video for reference and skip to 10:50:
Why is this so rare?

Such kind of printers usually harder to assembles, calibrate, and maintain because 3 axes machine is a bit more complex than 2 axes. For instance, it's can be tricky to move an entire extruder among all 3 axis and some of such printer's designs may require even dedicated exruder's design like Bowden Extruders.

Are there flaws in this design?

The key disadvantage of such kind designs is complexity with moving of an extruder among all 3 axes. Moving platform by at least one axis simplifies that.

Will print quality be affected by using this approach?

It depends on the exact printer's design, so, potentially you can have issues with ease of assembling and maintenance due to more complicated construction and as a consequence higher risk of low printing quality due design, assembly or configuration mistakes.
On the other hand, if you already have some device with precise enough 3 axis machine, like CNC milling machine, you can upgrade it to 3D printer by installing an extruder, however, it would also require update of software and, probably, electronics.