Why the fuck isn't there a place where we can see our ping in the client?

Because client ping and in game ping are different.

Because it would be very difficult to implement.

Your game is not assigned a server until the end of champ select. At which point you are assigned one of many servers for your game to take place on.

Since there is no way of knowing which server you will be placed on, there is no 100% way to know what your ping will be.

Because your ping can change based on a variety of factors, like your ISP shitting itself or computer being outdated for example. In client is easy mode coasting for most connections but once those in game particle effects start coming in then things get choppy.