Can’t transfer files from Macintosh HD-Data because of “broken seal”

Is this an Intel-based MacBook Air? Or an Apple Silicon based one (with a M1, M2, or M3 chip)?

If it is an Apple-Silicon-based one, and you have another Mac, you could try putting the MacBook Air into a mode where it acts like an external disk. See the "Use Share Disk to transfer files between two Mac computers" section of Use macOS Recovery on a Mac with Apple silicon - Apple Support . You would not need a high-end, expensive Thunderbolt cable for this – just a USB one (with appropriate connectors or adapters) that can carry data (i.e. not just a charging cable). I don't know if the broken seal on the cryptographically sealed system volume would cause a problem with treating your Mac as an external drive, but possibly in this case, it might not.

If it is an Intel-based one, I do not believe you will have that option in Recovery Mode. There is Target Disk Mode (Transfer files between two Mac computers using target disk mode - Apple Support), but in that case you would need to use a Thunderbolt connection, and I'm not sure it would even work given the state of your Mac's system.

I suspect that the “broken seal” is a complaint about the other part of the Macintosh HD - the cryptographically signed (“sealed”) volume that holds macOS files that do not need to be modified during normal operation. Drive corruption (which you say thot you are experiencing) could cause the cryptographic signature and the contents of the volume not to match up - telling the system that the volume had been corrupted somehow, whether through hardware or software failure or malware attack that had somehow slipped by the first layer of defenses. The Mac would then refuse to use the corrupted volume.

Use macOS Recovery on a Mac with Apple silicon - Apple Support

Use macOS Recovery on an Intel-based Mac - Apple Support

Another thing you could try is reinstalling macOS from macOS Recovery. See the "Reinstall macOS" section, of the appropriate Support article, where it says

"Under some circumstances, you may need to reinstall macOS. You can reinstall macOS while keeping your files and user settings intact."

Do not follow the ones in the "Erase and reinstall macOS" section.

Since your drive appears to be corrupted, I would assume that even attempting to do a Reinstall without an Erase might pose some risk of overwriting your data. But if you have no backups – and otherwise would lose the data – it might be worth a try.

I’m not sure that “Restore” means what you think it does.

”Time Machine System Restore: Restore your data from a Time Machine backup.”

I could be mistaken, but that doesn’t sound like a way to copy data off the Macintosh HD -Data volume to a USB flash drive. It sounds like a way to overwrite the contents of the Macintosh HD (including that volume) with data from a separate Time Machine backup. Assuming that you have such a backup in the first place.

Use macOS Recovery on a Mac with Apple silicon - Apple Support

Unless you made regular backups the answer is no, your data is gone.