It's convoluted because the requirement is to use offline and online machines and PSBT procedure.
That requires a watch-only wallet in the online Bitcoin Core and the actual wallet with private keys in the offline Bitcoin Core.

If you want simpler procedure (doesn't need to sync Bitcoin Core but w/ less privacy): You can use Electrum to create a watch-only wallet with "Import Bitcoin Addresses or Private Keys" option during wallet creation.
Paste your addresses there (one line per address) and finish setting up the [imported] Electrum wallet.
When using Send in Electrum, after clicking "Preview" or "Ok", it can export a compatible PSBT file in "Share->Save to file" that your offline Bitcoin Core can sign.

Transfer that to your offline machine where the 2013 wallet.dat is already restored in a Bitcoin Core version that has PSBT support.
Load it using the menu: "File->Load PSBT from file"; Sign it with "Sign Tx" (requires passphrase), then "Copy to Clipboard" for the next step.
The only additional step in Bitcoin Corehere is it needs to be converted from PSBT to serialized transaction so that Electrum can broadcast it:
So, open the console (Window->Console) and type: finalizepsbt "your_copied_signed_psbt"
Copy the output (e.g. to a text file)

And back to the online machine; transfer that text to your online watch-only Electrum via "Tool->Load transaction->From text".
Check the outputs, fee and amount if everything is good, click "Broadcast".

It's convoluted because the requirement is to use offline and online machines and PSBT procedure.
That requires a watch-only wallet in the online Bitcoin Core and the actual wallet with private keys in the offline Bitcoin Core.

If you want simpler procedure (doesn't need to sync Bitcoin Core): You can use Electrum to create a watch-only wallet with "Import Bitcoin Addresses or Private Keys" option during wallet creation.
Paste your addresses there (one line per address) and finish setting up the [imported] Electrum wallet.
When using Send in Electrum, after clicking "Preview" or "Ok", it can export a compatible PSBT file in "Share->Save to file" that your offline Bitcoin Core can sign.

Transfer that to your offline machine where the 2013 wallet.dat is already restored in Bitcoin Core version that has PSBT support.
Load it using the menu: "File->Load PSBT from file"; Sign it with "Sign Tx" (requires passphrase), then "Copy to Clipboard" for the next step.
The only additional step in Bitcoin Core is it needs to be converted from PSBT to serialized transaction so Electrum can broadcast it:
So, open the console (Window->Console) and type: finalizepsbt "your_copied_signed_psbt"
Copy the output (e.g. to a text file)

And back to the online machine; transfer that to your online watch-only Electrum via "Tool->Load transaction->From text".
Check the outputs, fee and amount if everything is good, click "Broadcast".