However, if you try to broadcast it through the gossip protocol, it would be dropped by any receiving node's mempool, except if they have some custom code that doesn't drop it and instead adds another input so it can be further forwarded towards a miner, I suppose.
But even then, you probably need to recompile your client to reach that. Or you could have some proxy that would capture invalid transactions and store some of them, but probably doing changes in Bitcoin Core and making a custom version that would store more transactions would be easier.
Exactly, it is true and you understand it correctly. I think everything was said, it is just a matter of reading some posts again. And maybe sharing a public key with vjudeu if you want to see that in practice (without losing any coins).
Edit: I wonder if people can combine transactions. Let's see:
0200000000010257c810eb18266e2a09e6f244c38f00731f6c1a29765dc3d1fd5142477101b2660100000000fdffffff8000da95a1986e5ebfe15fbf5fcc70864e12504fb87134a5f53fcc9ee461467c0000000000fdffffff01a0860100000000001600145df5785bcc56cd3079a798a9bdbbf45dd3217c00000247304402204482191bb6555308d065a8c7b9c24e617ee2e6202037c61a5a84f9b799e969ee0220456acaa25d8e794d7eca73be19980fa571e4a432b329b2c8183dbb9589b2d965012102fad16ad186269aa4caf89f19f690801b6f3fec89d04e1c4a40253f60e22eb87440d10c00
For now, it is locked, just to stick to the topic and make it really slow. But you can reply with your multisig, then I could make it faster, to form a complete example.