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This is all correct, but there are privacy implications to consider.
One may have UTXOs in sha256-bitcoin that are valuable enough to not consolidate for privacy reasons, that is, where the privacy value of keeping them unconsolidated outweighs any benefits that would accrue from doing so. They may perhaps be valuable enough to just spend/trade some day one by one.
Then you take the same UTXOs on blake2b-bitcoin, they have much lower market value and you end up consolidating them. This unfortunately allows a sophisticated multi-chain analysis entities to take information given on bitcoin-blake2b and apply it for ownership heuristics on bitcoin sha256.
I wonder how many OP_RETURNS are out there that show intention to fork coins and what further inferences can be made from them. A cluster of "random" data ones in a short time frame may be reason enough to suspect some corellation.
Very good advice about using electrum to cache a movement transaction first. If one forgets to do that, I came to realize that electrum has a way to make a synthetic history: take it offline (invalid localhost server will do), load a fresh wallet and there will be no transaction history. A desirable history of relevant UTXOs can be loaded via tools->load from *, and the popups have a add to history button.
It is tricky to know for sure that a UTXO that seems to be on bitcoin-sha256 only (origin story not known) will stay that way. A send to self at least creates a checkpoint, a competing transaction that one expects to use on blake2b-bitcoin only will at least not relay on that network if it turns out the competing send to self sent first relayed on both networks.
Another bit of electrum fun, the movement of a known bitcoin-sha256 only UTXO from one wallet can be paired up with self-movement of another UTXO from another wallet by way of coin join features done before any signing. This doesn't provide any coinjoin privacy (the one to one movements are obvious), but it does help normalize coinjoin and reduce a wallet to wallet transfer step.
Here's a safer procedure that doesn't involve running shady wallets: