segregated witness: How can Segwit increase the performance of the transaction if the same amount of data is stored in the block chain?

I am publishing this after reading the answers to this: Where is the signature stored in a SegWit transaction?

I found out from reading that, that in Segwit, the "witnesses" (or signatures) that verify the authenticity of a transaction are still stored in the blockchain. In that case, a given transaction must use the same number of bits as if Segwit was not being used. If the block size remains the same, say 1 MB, then it should contain exactly the same number of transactions.

I have an alternative possibility. Let's say that at present sent to the nodes is a transaction with an empty signature field (I think that field does not exist due to multiple signature things, but be patient with me). Then, the signatures can be sent through a separate channel. What at present is stamped on the blockchain is a transaction with an empty signature field. And then, due to the longer blockchain-always-wins rule, nodes that are not Segwit will simply accept that the blockchain with the most transactions, which includes some funny transactions without a signature, is the authorized blockchain.

But apparently the reality is more complicated.

Also, there is a question about how much you can abuse the rule the longest block chain always wins. (The word "abused" does not imply a negative opinion). For example, let's say I upload with SegFoobar. SegFoobar is the same as Bitcoin but with a larger block size (without Segwit). Once enough nodes adopt SegFoobar, then more nodes will win the mining race, so other nodes will effectively accept larger blocks because the history is longer. I know that's not the case, but why?