Wow! This thread is timely for me. We are in the middle of planning a 3 season room off the back of our house and I WAS thinpking I wanted French doors but I was trying to find doors that would open out instead of in to the main room of the house. My goal was to allow an air flow between the rooms, except in the winter. One side would open into a corner and the other would take up some space but we could live with just one door open. I didn't consider sliders even though my in-laws love their sliders with blinds inside and screen doors. I am a bit worried about security and sliders. I didn't consider dirt moving with French doors nor the absence of screens which may or may not be an issue here. We have to move heat ducts for our construction so we are also considering everything possible for structural issues. I'll be watching as you guys talk. Thanks!
We did something similar a couple of years ago, but it's a 4 season sunroom.
We have both french doors (going from kitchen to sunroom) and a slider (going outside). See pic below - slider is next to chair on the left. We previously had just a slider going out to a deck where the sunroom is now.
I love my french doors and I love the slider, and I love them for different reasons.
We went with a french door instead of a slider here because, when open, the two areas meld together better than they would have with a slider. The french door is a smaller one and was made by the lumberyard where my builder does business. It is very nice quality and operates smoothly and quitely, has a very tight seal, etc. We used an outdoor door since we planned to keep the sunroom closed, without heat sometimes, so the outdoor door provides more protection from the cold to the inside of the house than the indoor french door would have and is still very beautiful. (I don't think most people notice the difference.) Good choice.

(We did not bring the ductwork into the new room but we do have electric heat out there that we rarely have to use except maybe on the coldest days and/or if the doors are shut overnight, for first thing in the morning.)
Naturally the outdoor doors were more expensive than the indoor door, and we got just the grate on the inside so the doors are a dream to clean. We decided against the blind essentially because we didn't need it as we have nice shades that we can lower for privacy if desired on the windows in the sunroom. The french door opening does interfere with the trash can (which is behind it when open) and the chair at the far end of our table (if someone's sitting in it), but those were things I decided I could live with in order to have it. It's not a huge issue most of the time. We've adapted by keeping a small bag for trash elsewhere in the kitchen and if we have company, I put the trash can in front of the door so the chair being in the way isn't a problem. (Gotta love living in a small house!) And oh yes, we also had already finished renovating the kitchen when we (I

) changed our minds about this so the wall had to be built up a little (from a 6 ft slider to a 5ft french door) so the cabinets and counter ended a little too short of where the wall ends, but again, we decided we could live with that and it's really not noticeable for the most part to anyone but us, as it turns out we keep the doors open most of the time. (Moral of the story: try to decide for sure ahead of time!)
The slider going outside is an older model but brand new Anderson slider. It has the wooden grates that come off to clean the door, and a separate screen door. I love the door as it's beautiful and strong and smooth, but honestly taking the grate off to clean the door is a PIA, it takes more work than you'd expect and I feel like it's going to break (and I have to clean it a lot as it gets dog nose drips on it). We got an excellent price on the door but if I had it to do over again I think I might have sprung for the inside grate type door, but then again we had a large renovation going and it was important to save money where we could so maybe not. But just things to think about. It does feel like a very safe door and has both a lock above the handle and a foot type lock on the bottom of the door so that it can't be forced open easily. (I am aware that sliders can be taken off their track so maybe it's a moot point, but with french doors all one has to do is break the window and reach in and unlock the door, so not so sure that's much better from a safety standpoint either.)
PS We have 8 Harvey windows in this room also and they all have the grates on the inside. Beautiful, and easy one-hand operation. I really love them!