Hmmmm .... not a silly idea. I used a 1D III as my primary birding camera for some years, then replaced it with a 1D IV. They worked very well indeed, offering all of the advantages you have outlined in the OP - very fast, brilliant AF systems, good noise performance by the standards of the day, and extraordinary toughness and reliability, coupled with a huge battery that lasts for several days of intensive use.
I used them with a 500/4 for the most part. None of the 1D cameras have ever had very good pixel density. (The 1Ds units did, but they were a different series.) The 1.3 crop helps a bit, of course, nevertheless, they don't put nearly as many pixels on the bird as (for example) a 70D, a 7D Mark 1 or a 5D IV. With a 500/4, that was OK. I'm not sure that I'd really want to use a 1D as a first-choice birding camera using a 100-400. You'd struggle for reach a bit too often.
I found the ruggedness and all-round class of the 1Ds, allied with their pretty fair low light ability, made them an excellent choice as a birding camera. The relative lack of reach was something I could live with (being blessed with a 500/4), especially after I added a 7D Mark 1. From that time on I use the 7D in good light when reach-limited, and the 1D IV for everything else. It was a great combo, and the 1D IV remained my most-used camera for many years - not just for birding, I loved using it for landscapes too.
Eventually, the 1D IV got pushed out of prime-time use by a 7D II and especially by the 5D IV. It is still a joy to use - it's the most responsive and all-round fun camera to use I have ever owned - but I get tired of the sheer bulk and weight of it. I habitually carry two or three cameras, and these days I tend to take an EOS R as the third body, simply because it is small and light. Nevertheless, I pull the old Mark IV out quite often, and always find it delightful in the hand. Are the shots sub-standard compared to the 30MP full-frame goodness of the 5D IV or EOS R? Not really. It has always had wonderful colour straight out of the camera, and looking at a big print, you very seldom remember or notice that it was taken with a "low" resolution body.
Would I select a used 1D III for birding today, if I was on a budget? No. Not when you could get a Mark IV, which is a noticeable step up.
What about a Mark IV? Maybe, but it depends on my intentions. If I was going to do a lot of reach-challenged stuff with lighting not a problem (waders on a beach, for example), then I'd go for a 7D Mark 1. Excellent reach, responsive, good to use. The only issue is that 400 ISO is your limit for really clean results, 800 ISO if pushed. It's not really usable past 800 ISO.
If, on the other hand, I was doing a more varied selection of tasks, the Mark IV would be an excellent choice. And if I was doing a range of wild subjects in a range of environmental conditions - from small birds right through to large mammals, then the Mark IV would be ideal.