I am aware that the current viewkey setup allows us to check for funds sent to our wallets. However, in the example case of cold storage, it would be very nice to be able to check if funds are still there as well, giving us a sort of "live" view of a wallet balance. Is there a possibility of the viewkey giving us that ability at some point?
Very unlikely, unless the crypto changes substantially.
However, if your goal is to see whether your cold storage has gone bye bye, there is a way to do this (though no current tooling to allow you to do this easily currently): the cold wallet can export the key images for the outputs it received, and a hot wallet can watch the blockchain for those. If they appear at some point, it means the corresponding output was spent. Hopefully by you :)
Feel free to file a bug on https://github.com/monero-project/bitmonero/issues asking for that particular capability to be added to the tooling.
As you stated, the view key can only be used to view incoming transactions.
The spend key is needed to view outgoing transactions. Using the spend key is not currently possible with a cold wallet (without exposing it to the Internet).
There us hope that offline hardware wallets such as Trezor will be able to solve this problem eventually.