Create a catio for them with their own entry point from a window or cat door. When they have a safe enclosed area to go outside, they are far less likely to attempt to mastermind their way through the people entrance.

You may want to just keep the sliders and add screens if you want the cats to stay inside. I have Pella sliders with a screen that's on the interior side of the sliding door and I consider it perfect for use with pets. It can be latched/locked so there's no chance my cat can get outside.

Since your cats are indoor/outdoor now, consider doing what holly mentioned. My cat has a cat door in the BR that opens to a wire tunnel. The tunnel then leads to a 10', six ft. high kennel that sits in the corner of the deck. Makes for a happy cat..........

Thanks all. I think the catio will be the best option since I'm fixated on losing the sliding doors. Now I just need to figure out where to put it and how big it should be. Hopefully, this solution will satisfy the kids.

I would still look into a framed screen since your cats are going to look for any and all exits.
Even my cat will sometimes hide near the door and then try to dash out behind my dog. A squirt with a water bottle usually reminds her not to hang out around the doors. : )
What she loves to do when we go out the sliding doors is run to her cat door and try to beat us outside. Have to say that having a covered deck made setting up her outdoor area much easier...........

Definitely get framed screens. Even with a catio, cats will try to investigate where they should not be... and as someone who grew up in the country and had 2 cats hit by cars despite living on a country road set far back from it, I can say cats are no safer outdoors in the country than in the city.

You could perhaps get swing-out framed screens like a screen door on a front door?

Swing out framed screens with swing out french doors? I'd love to see how that works. It seems that it would be the best solution but I've never seen both out swing doors and screens. Anybody?

Nope can't have both outswing. Inside hinged are available but it defeats the purpose of the outswing and eliminating the space required for the panels to swing in.

Thanks, Millworkman. Just as I thought.

Tinan; the issue in question is inswing vs outswing. One set must swing inward and the other outward.

Thanks to all!