How can Segwit increase transaction throughput if the same amount of data is stored in the blockchain?
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.