"for Non-custodial wallets, it is required you to store your private keys, as it gives you sole responsibility over sensitive data."
First, are private keys for Non-custodial wallets to be looked out for and stored/saved just like the seed phrases?
Yes private keys are as important as seed phrase. Because one with private keys coyld easily get access to your wallet account for example, i have a scenario for you. When we make metamask wallet as extension on chrome or any browser. We then asked for password to log in but if we have forgot the password we are asked for private keys to login to wallet (to recover the wallet) if you don't have the private key then you have to uninstall the wallet and then during creation you have to select the import option and then put the seed phrase or you will see another option of recover from private keys.
So, i hope you know might understand the importance of private keys.
Secondly, what other times would a wallet owner need to personally use the private keys aside from its primary automated role of authorize transactions?
I have already answered this query in the last answer but instead of recovering accounts private keys are used to authorize each activity on your wallet like making swap, transaction etc. But that depends on the wallet you are using as not every wallet will ask for Private keys as another wallet might be asking.