Satisfying a cat's need for the outdoors with 2x daily walks?

I'm trying again to make the switch to indoors only for my indoor/outdoor kitty.

We live in a very rural area and he loves being outside chasing rabbits through the fields, climbing trees, etc.

He will often claw the windows and doors, look for escape routes, etc. when he's been indoors for awhile.

I was wondering if anybody has been in the same situation and has managed to satisfy the kitty by switching them to indoors only, but with daily or 2x daily supervised walks. I have used a leash and harness with the kitty and he tolerates it fine. 

Anyway, wish me luck as I make the switch! I have 3 young kids and the biggest challenge is to keep them from leaving doors open. I explained to them that they need to all be on board with me to make it work. 

Is he fixed? My once-stray Sammy did the same thing (clawing at screens, etc) until we had him neutered.

Yes, he is neutered.

We live in an apartment so our kitties have no choice but to be indoor kitties.  Snowy is a recue who was found out at a lake, nearly starved to death and with mange so bad she had lost most of her fur.  She is also deaf.  Tati is a ragdoll we got from another lady who could not keep her because of Landlord issues (or so she says... this kitty is very high energy and very nocturnal and I think that is more what the problem was).  Tati's previous owner said she was strictly an indoor kitty.  Tati is the one who does best on the harness and needs daily walks to expend energy.  She climbs trees, chases bugs, we even have some luck with leading her so we actually cover some ground while we are walking.  She loves the outdoors.  She even made one escape from our second story balcony and gave us a good scare that we had lost her!  I walk her every evening, and she knows I am going to have a cup of coffee and a break when I get home from work, but if I relax too long, she goes nuts.  Snowy on the other hand is hesitant to go out.  She likes the safety of the stroller better than a harness She  likes going to Petco better than going to the park.  When she is on the harness, it takes her a long time to get into explore mode.  About the time I am ready come in, she is just getting started.  Her big thing is sniffing every inch of everything in her path.  I tell my daughter its because she has been lost before and wants to make sure that she can find her way home, but cats learn so much about the neighborhood by sniffing a single blade of grass.  Snowy is also allergic to fleas so she hasn't been crazy about going out this summer.  The fleas are bad here. Anyway, if she goes out, it usually means I coax her out, and then I have to pick her up hissing and fighting to get her back inside when its time.

I think it depends on the cat.