Twitter Card works in Card Validator but not when sharing the page unless it has been validated first

I'm trying to make some news pages to show a Twitter Card with an image (summary_large_image) when shared on Twitter. I've added the necessary meta tags on the page. The validator says everything is fine, and actually, when I share the page after validating it, it appears fine on Twitter as well.

The problem is that if I share another page, without going in the Card Validator first, the card doesn't appear on Twitter. If I go to the Card Validator afterwards with the page URL, it will make the card appear both in previously shared tweet, and in new tweets.

Shared pages won't have a card even after tens of hours, unless I validate them with the Card Validator. I obviously can't expect regular users to copy/paste the URL in the Card Validator each time they want to share a page, so I must be missing something.

Here is an example tweet that doesn't work: https://twitter.com/BertrandHilaire/status/857855909447569408

Here is how it renders at the time I'm writing this question :

Notice that it might start working if someone validates the URL in the validator.

Here is the output from the validator (for another page):

*.etoilenotredame.org is whitelisted for summary_large_image card 

And in the Log area:

INFO:  Page fetched successfully
INFO:  15 metatags were found
INFO:  twitter:card = summary_large_image tag found
INFO:  Card loaded successfully

Thanks for inputs, I'm out of ideas!

Apparently, I just needed to wait a little more. Made a few tests today (4 days after implementing the Twitter meta tags), and everything works fine. The first tweets I made have not been updated, but new ones correctly display the card, without delay. Maybe Twitter needs time to register that a website has implemented the card system.