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

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well the issue with ping is like doing a ohm test on a wire. you can put your meter on both end of the wire and see little to no resistance but that dont tell you that wire can carry a load of current efficiently. you need to load test the circuit/wire to make sure the wire/circuit can handle the load.

so seeing you ping in client outside of a game is kinda the same. there is very little data transfer/packets going back n forth but when you load in game it has to transfer all the data for all 10 players plus server to process the game correctly. that would be a load test.

Load is pretty negligible. Every action boils down to Player # did Action targeting Target, which can be represented in about 4 bytes or less (5 bits for Player ID, 1 byte max for Action Type and any required info, 2 bytes max if the target is an X, Y location, possibly less. Still round up to 4 bytes for this example). Even if you assume a teamfight scenario where everyone is performing at 300 APM you still only hit 200 bytes per second, or 1.6 kbit/s. For reference, dial up internet caps at 56 kbit/s. Lets just say there's a reason Riot doesn't bother listing a minimum internet speed for playing the game.

also there are desktop apps you can use to see ping at all times, and websites to test your ping.

Those desktop apps ping to whatever their respected websites are, not to Riot's game servers. You could very easily get a drastically different response to one of those than you would to Riot, especially if they're testing your ping to an ISP.

Besides, "a clunky work around exists" isn't a good reason not to include a feature.