I don't think X.509 has a way to express multiple signatures on the same certificate. Issuer name, etc, are limited to one per certificate. See also tialaramex's excellent sibling comment about CAs constructing a new cert to sign including a new serial number.
Cross signed roots work by offering multiple certificates for the same key: you can use a self-signed root (from your trust store) or an intermediate signed by a different issuer, or if your validating stack is really competent, you can send multiple intermediates and the validator will check if any of them chain to an acceptable root.
But, the leaf (or end-entity) cert MUST be the first certificate sent, and only one certificate can be first, so there's no optionality there.
If CAs would be willing to sign limited scope intermediates (and if limited scope intermediates were widely usable), you could get your intermediate widely signed and have your leaf certificate signed by that, and include multiple chains from your intermediate. But that would take you from two certs (leaf + CA intermediate) to one + 2N certs (leaf + (your intermediate signed by CA + CA intermediate) * each CA) and all of that would add up to increase the handshake size and slow down initial communication.
It might be nice in some situations, but it's also costly, and support is iffy if you stray outside browsers.
Cross signed roots work by offering multiple certificates for the same key: you can use a self-signed root (from your trust store) or an intermediate signed by a different issuer, or if your validating stack is really competent, you can send multiple intermediates and the validator will check if any of them chain to an acceptable root.
But, the leaf (or end-entity) cert MUST be the first certificate sent, and only one certificate can be first, so there's no optionality there.
If CAs would be willing to sign limited scope intermediates (and if limited scope intermediates were widely usable), you could get your intermediate widely signed and have your leaf certificate signed by that, and include multiple chains from your intermediate. But that would take you from two certs (leaf + CA intermediate) to one + 2N certs (leaf + (your intermediate signed by CA + CA intermediate) * each CA) and all of that would add up to increase the handshake size and slow down initial communication.
It might be nice in some situations, but it's also costly, and support is iffy if you stray outside browsers.