Cards don’t load in TweetDeck - they work in iOS, Android, and Twitter.com. It’s a common request for the TweetDeck team, I’m sitting right next to them as I type this! It shouldn’t matter if you use TweetDeck to schedule and post a Tweet though - the card will render in the final Tweet in the clients that support that display.
You should only need to check your site in the validator once for the domain to become whitelisted, after which our crawler will know to check for your markup and render the card.
Overall, we can do a much better job of explaining this in the documentation to reduce these common confusions, so I’ll make an effort to get involved in that the next time we are updating it!