Why do we need public key as part of Witness in P2WPKH transactions?

Why do we need public key as part of Witness in P2WPKH transactions?

As we want to minimize the data payload in a blockchain, which also led to concepts like SegWit, and is matter of discussion in Signature aggregation etc, I wonder why do we actually need to supply public key in order to spend? It's needless 33B of data. ScriptPubKey could look like this <HASH160> OP_CHECKSIG2 and ScriptSig <Signature>. I know this requires Softfork, but as we introduced SegWit, why wasn't this the design? Is it because mapping signature -> public key is not unambiguous?

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