the DLEQ is more for the software wallet to validate that the hardware signer a) didn't fuck up (bitflip) and b) isn't malicious (sent the funds to salvatore!)

a nice property of SPs is if the address is known to the hardware wallet as 'josie', the hardware wallet is fully..

.. responsible for generating the on-chain output, so there really is no way for the sw wallet to trick it. this gets really cool with BIP353 where the software wallet can ask the hardware wallet to send to josie@josie.gg and if the hw can do a dns lookup, it can get the DNSSEC..

.. SP address, and then generate the correct on-chain address, while showing to the user it did in fact get the right SP address for "josie". if the hw wallet is not internet connected, the user can register the SP address for josie (think "add a contact") and from that point ..

.. onwards the software can say "send to josie" and the hw wallet can use the sp address registered on the device to generate the correct on-chain address without the software wallet being involved

tldr; SPs are truly "push payments" in that whatever device is doing the cryptography (the hw wallet) is the one fully responsible for generating the output. the DLEQ is to keep the hw wallet honest by requiring a proof that the output in the signed tx is the correct one