How can Segwit increase transaction throughput if the same amount of data is stored in the blockchain? I'm posting this after reading the answers to this: Where is the Signature stored in a SegWit transaction?

I've found out from reading that, that in Segwit, the "witnesses" (or signatures) that verify a transaction's authenticity are still stored on the blockchain. In which case, the same amount of bits ought to be used up by a given transaction as if Segwit weren't being used. If the block size remains the same, let's say 1MB, then it ought to hold the exact same number of transactions.

I've got an alternative possibility. Let's say that what actually gets sent to the nodes is a transaction with an empty signatures field (I think that such a field doesn't quite exist because of multi-sig stuff and scripts, but bear with me). Then the signatures can be sent through a separate channel. What actually gets stamped onto the blockchain is a transaction with an empty signatures field. And then because of the longest-blockchain-always-wins rule, the non-Segwit nodes will simply accept that the blockchain with the most number of transactions, which includes some funny signature-less transactions, is the authoritative blockchain.

But apparently reality is more complicated.

Also, there's a question about how much the longest-blockchain-always-wins rule can be abused. (The word "abused" does not imply a negative opinion). For instance, let's say I come up with FoobarWit. FoobarWit is the same as Bitcoin but with a larger blocksize (no Segwit). Once enough nodes adopt FoobarWit, then more such nodes will win the mining race, and so other nodes will effectively accept bigger blocks because the history is longer. I know that's not the case, but why?
The answer to the question "Are the segwit witnesses part of the blockchain" depends on what you define as the blockchain:


According to old pre-segwit nodes, the answer is no, as they don't care or receive the witnesses.
According to new segwit nodes, the answer is yes; the witnesses are as much part of the chain as everything else, and subject to just as many validation rules.


So segwit increases throughput slightly, at the cost of increasing the size of the blockchain. There is no magic bullet here, and this aspect of segwit is simply a block size increase. It is not a scalability improvement.

Segwit was primarily a solution for the transaction malleability problem, opening the door for various higher-level protocols that were handicapped by being unable to reason about future on-chain transactions that weren't published yet. In addition it also improved he incentive structure a bit (making spending relatively cheaper w.r.t. creating outputs). As a final side effect of that, it also increased the block size a bit in a backward compatible way.

As for your question about another fork that goes even further: what your software does has no effect on those who don't adopt it. Even if it has all the hashrate in the world on board, old nodes won't care or see your additional data.