This would be helpful! Unfortunately, there is no built-in way to do this in Airtable, because it doesn’t remember your sort order from one view to the next.
As you mentioned, you would have to create your own additional field to keep track of the sort order numbering.
It would be nice if the Batch Update block enabled us to increment numbers throughout a list of records, but it doesn’t allow that.
Your idea plus these ideas would make for good feature requests, if you want to add any of them into the #show-and-tell:product-suggestions category.