Old laptop died during migration to new MacBook Air

A short explanation for that: the installer performs a somewhat cursory inspection of the drive on which it's about to install macOS. If it fails, the installer tells you to "repair" the startup disk with Disk Utility as it did. I write "repair" because whatever it needs to do shouldn't have been necessary to begin with, and whatever "repairs" First Aid performs are likely to be temporary.

Basically... not a good sign. What you can do is quit Recovery mode and restart your Mac in single-user mode: Start up your Mac in single-user mode - Apple Support.

You'll log in as usual, and then you'll be confronted with a wall of very tiny text. Eventually it will finish scrolling and a prompt localhost:/ root# will appear.

At that localhost:/ root# prompt type the following

fsck -y

... followed by the Return or Enter key.

You might be rudely interrupted by some task that had been working in the background, so if that happens just hit the Return or Enter key again to summon the localhost:/ root# prompt again.

That fsck command isn't magic. It performs the same cursory disk inspection and "repair" as Disk Utility, and is just as likely to fail. So why do it. It's been my experience that repeated and persistent attempts at using that command have been known to result in a successful "repair", long enough to start the Mac and attempt to back up its data. To repeat the command you can hit the "up arrow" key which saves you from having to type it again.

If you're really lucky it will eventually end with

** The volume ... appears to be OK.

I don't want to give you false hope though. Those repeated attempts that eventually succeeded were with spinning hard disk drives. To date I have not found it to succeed with flash memory / solid state drives due to the way they tend to fail. You have nothing to lose though.

To finally exit Single User mode type

exit

... and the Mac will continue to start up (if it can).

Reinstalling macOS is the right thing to do.

That action does not erase any of your User Account information or data—however—since the Mac "died" that data may already be gone. Flash memory can go all at once, and the symptoms you describe suggest prospects for recovery are dim.

You won't know anything until you try.

I prefer to use Time Machine for a variety of reasons, but there is no effective difference in the way data transfer occurs. If you're successful in getting that Mac to start the first thing I'd do is to back it up with Time Machine. It's simply not possible to speculate the likelihood of success.

Am I better off waiting until I can get to an Apple store?

That would be ideal, if you have the patience. They're almost certain to come up with different ideas or suggestions. It's also possible to physically remove that Mac's solid state memory and install it in a separate enclosure so that it's more easily examined or "repaired", but I'm really skeptical about chances for success. The correct and only certain way to fix a corrupted B-tree is to reformat the drive... which erases it.

It seems the old one gave out at the worst possible time. No Time Machine backup?

See if you can get it to start up: If your Mac doesn't start up all the way - Apple Support

If your Mac starts up to an Apple logo or progress bar would be the most applicable. Exhaust each one of those steps. For example you might be able to start it in Safe Mode. If you can, back it up. Then you'll have options should it fail to start again.