I prefer the Datejust on the original bracelet, I don't think the design works well on a strap although it looks better if some type of end link is used. The case isn't finished between the lugs because it isn't supposed to be visible, and for me that always looks wrong. I also think the watch starts to look a bit tall on a strap.
The pre -2006 Datejusts have very slim lugs and on the bracelet the watch almost looks like an integral bracelet design; the lugs almost flow into the bracelet because they're so tapered. This effect is more noticable on a larger wrist too, it's a subtle point that's not obvious till you wear one.
I own 2 Datejusts and I could get tempted to buy another. Both mine are fluted bezel/Jubilee bracelet models (one's steel, one's bimetal) but I've always liked the bimetal Oyster bracelet/smooth bezel variants and if I found a real minter I'd probably buy it. You can't have too many 36mm Datejusts (or vintage Omegas) IMO.
If I was the OP I wouldn't spend a lot on a strap, try it on a cheap one to see if you really like it. Each to his own, but I'd stick with the bracelet.
Paul