So it looks like I need the enter the seed no matter what, right? That sucks!
If you have your wallet seed, it works fine. If you don't have it because you did not back it up, it sucks actually!
Is there really no way I can update the Google Authenticator app, then install the latest version of Electrum, place my wallet.dat file on my computer, and then use the Google Authenticator 2FA token to send the money
The bottom line is you can use your wallet seed, again if you have it in your backup, now you can import it in your Electrum wallet and get a new wallet, without 2FA.
With same wallet seed, same private keys and same public addresses like your 2FA wallet.
[GUIDE] How to Safely Download and Verify ElectrumThe 2FA in your Electrum wallet provided by Trusted coin.
Electrum 2FA walletTwo Factor Authentication
Electrum offers two-factor authenticated wallets, with a remote server acting to co-sign transactions, adding another level of security in the event of your computer being compromised.
The remote server in question is a service offered by TrustedCoin. Here is a guide on how it works.
How to recover access to the bitcoins in your 2fa wallet.