You can boot to a CD to flash the BIOS, or even boot to the hard drive if you make a small DOS partition. A floppy drive is just the simplest, most basic way of doing it. You can possibly even do it with a USB memory stick, but it's a bit of a hassle to make a memory stick bootable to DOS. Many mainboard makers even provide Windows-based flash programs now.
It's often considered a more "reliable" way of doing it too, but in fact, the floppy controller isn't even part of the system chipset, it's on the LPC chip, so that's one more device that might fail in the middle of a flash, compared to using an integrated hard drive controller. Not to mention the inherent crappiness of floppy disks. But floppies are portable, cheap, you can usually find a computer to write a floppy with.
Floppies are just a "least common denominator" in terms of how a mainboard maker specifies that you flash the BIOS. Nearly everyone has one, and if you don't, you probably can figure out how to do it without a floppy.
As for gameports, they were slow, chunky, taking up valuable space and used an IRQ, and only a small percentage of people used them. With the advent of USB, and people rushing to get newer joysticks with a faster interface, and "digital" performance (and I don't think the gameport was designed for hotplugging technically), there were even fewer people looking to use a gameport. They lasted a few years, and even still are available on soundcards or at least via a connector cable to some. The number of people desiring one just gets less and less as time goes on. Most devices that use them ought to be worn out or just too grungy to use anyway.