What does "k" in the constant product AMM formula actually represent?

I've been sitting trying to figure out how somebody would start with the requirement of an automated market maker to always be capable of providing a price quote for either the buy side and the sell side on any token pair, and then be led to the constant product formula of x*y=k, or really why a market maker's pricing function for a given order has to be a constant to begin with.

What does k represent? The answer I read is "liquidity" or "the value of the pool" -- but I would argue that the value of the pool is much more accurately represented by (Amount of token A) * (Price of token A).

I've read Vitalik's original post, but still not seeing where the original idea of this specific product formula comes from, or what the invariant truly represents. Prediction markets seem much more akin to the StableSwap formula of a constant sum (which equals 1 in the case of a PM).

x*y = k , is basically saying that our metric is measuring every whole token A as the entirety of token B's supply in the pool. Why would you do this?

The formula itself produces some quadratic pricing effects that seem far less than desirable, too -- I understand that Curve has improved on this in various ways, but what was the motivation for starting this way?

Thanks for any answers you can provide.