Bitroot

I have a security question that has been bugging me.
When a wallet broadcasts a Bitcoin transaction, we trust it's only sending the signature and transaction data. But how can we be certain that fragments of the private key aren't being secretly embedded in the broadcast over time?
For example, could a malicious hardware wallet manufacturer design a device that, after many transactions, allows them to reassemble the bits and know the private key?
Has anyone ever done a public test where the same seed phrase is used on different hardware wallets (like @nprofile…22gk @nprofile…vysa, Trezor @nprofile…q32z, @nprofile…enut) to sign the exact same transaction?
If the resulting signatures are identical, would that be definitive proof that both devices are performing the standard, non-corrupted signing process?