First of all, I’m really sorry that you’ve hit this issue.
It looks like the player on your website is using Flash, so when one of the URLs is Tweeted, the twitter:card player tags in your web pages point to a Flash object.
We announced that we would sandbox (no longer be supporting) these kinds of players for new partners in August last year. In April we announced that we would discontinue support for Flash in any of our cards, and switched off support in June.
Moving to HTML5 and more modern technologies means that we are able to provide a better experience across the board on both web and mobile platforms.
There are some suggestions for how to migrate your players to more modern technologies in our announcement post, which is the third link above. We also have a sample player card shared in our Github:
Apologies for any gap in communication here that caused you to miss the update, we do work hard to provide notice with changes like these, and posted as much information as we could over an extended period to support cases like this.