About 10% of the population is left-handed. Tua Tagovailoa is the only left-handed quarterback in the NFL. He's not just the only left-handed quarterback in the NFL with a starting job; he's the only quarterback, starter or backup, who throws with his left arm. What's more, in the last seventy years, only thirty-three left-handed quarterbacks have made NFL appearances.
So, I was curious and dug up some reasons why and, yes, discrimination appears to be a factor. Steve Young was moved to safety his first year at BYU because his offensive coordinator wouldn't coach southpaws. Fortunately, the offensive coordinator left and Young was moved to quarterback the next season. John Gruden told Chris Simms in no uncertain terms that he wouldn't see the field much because Gruden didn't want to flip his plays to accommodate Simms' left arm.
That last part is a factor. Most plays are drawn with right-handed quarterbacks in mind. It's easier for right-handed quarterbacks to roll out to the right and throw to the right side of the field. For left-handed quarterbacks, everything is flipped. Throwing across your body on the move is a hard thing to do and even a lot of NFL quarterbacks aren't good at it. So, plays that are drawn with the intended reciever on the right either have to be adjusted for a left-handed quarterback or the quarterback has to make a harder throw.
The spiral of the ball is also opposite when thrown by a left-handed quarterback which can be a difficult adjustment for recievers. When the 49ers moved on from right-handed Joe Montana to left-handed Steve Young they actually told much of their staff to learn how to throw left-handed so the recievers would have the same spiral during warmups as they'd have during the game. This can also make it difficult for teams to select left-handed quarterbacks as backups because, if your right-handed starting quarterback gets hurt, you have to make the adjustments mid-game.
As a former lineman, I can also say that there can be problems with blocking. The "blindside" tackle has become synonymous with the left tackle position because most quarterbacks are right-handed. If you have a right-handed quarterback and the right tackle blows a blocking assignment, the quarterback can see the pressure coming. If the left tackle blows his block, the right-handed quarterback will get, well, blind sided. So, most of the best tackles have been playing on the left since high school. Madden isn't real life; so, it's not all that easy to just plug the left tackle over on the right, especially with the complexity of NFL blocking schemes. Regardless, switching from a right to a left-handed quarterback immediately alters blocking assignments at least slightly.
Still, I think that Tua Tagovailoa had one of the best explanations for the dearth of left-handed quarterbacks: most lefties with a strong arm decide to play baseball.