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This chap used a hand-held router (and warns about how it likes to tip over mid-cut).
His design isn't quite what I'd go for, and I think your overview is pretty close.
I've wanted an excuse to get a biscuit jointer thing for ages, which I might use in preference to tongue-and-groove for everything.
Agreed on the seasoned wood, you'd be surprised at the lack of sawmills in SE23 though, sadly.
The chap in the link considered the mortice and tenon situation and decided on floating tenons, which if I understand things correctly is basically a big biscuit joint.
My 2p worth on the door chat.
I've made smaller panel doors and they don't half like to warp. I suspect those original doors are made from slow-grown pitch pine. Modern Scandinavian pine is absolutely shit in comparison. You may get away with poplar.
So get friendly with a good sawmill because I suspect you will need good, properly seasoned wood. Teak maybe. Acacia might do a nice job if you can get it. It works quite well for outdoor furniture if properly protected from the elements. Take advice from the sawmill.
The other reason to get it from a sawmill is the cost. Much cheaper than buying retail.
Using hardwood make the doors extremely heavy compared with pine. So they will be hard to hang and may put too much stress on the frame.
Depends on the sawmill but if they can't supply it in tongue and groove you'll need to do that yourself, and the best way is with a router table.
The construction is probably like a barn door with big thick sides&7 top and the lower part is Z shaped frame on the inside to nail the T&G planks onto. The horizontals are probably mortice and tenon joints. I've never tried using them because it's too hard and I don't have the right tools. Maybe a router table and a table saw will do it
The sides and visible horizontals are tongued or grooved to accept the planks too.
A router table would be the best tool for making the glazing bars too, but I've no idea how to construct that. I think there is one horizontal and 4 verticals. However, I'd caution against including the glass. There's no fucking way I'd have windows into my garage, especially not in South London, so think about panelling that and just sticking on some routed sections to reproduce the pattern.