Based on Blue Streak's catalog illustration and the two dated Copake bikes, I'm sure your bike is from 1897 or 1898. That means the 1902-03 New Departure brake must be a later upgrade, which would not be unusual. By 1902, five years later, the overall bike would have been very different; technology and style were changing very quickly. So I don't think the bike could possibly be that new. Interestingly, while the 1898 catalog describes an updated rear brake with "improved bearings and detacheable clip," which sounds like a coaster brake, the illustration still shows a spoon brake. It could be there was some overlap.
It sounds like you want to ride the bike. For seats I'd suggest a matched pair of new Brooks stretched leather touring saddles. While they are not the same as antique American stitched-and-padded leather saddles, they will look the part, are readily available, and are comfortable. Genuine antique saddles will be either A) very expensive if in good original condition, and not suitable for riding because of the brittle leather, or B) needing recovering, which can be done but is not cheap. There are folks who can authentically restore and re-cover antique saddles, but think many, many hundreds per saddle.
Tires are tough. As you say, whether single-tube or Morgan & Wright double-tube, the originals were probably glued-on tires. A few TOC wood-rim bikes used regular Dunlop tires (what cyclists now call "clinchers") or real G&J antique clinchers (unknown today), but as far as I know there's no modern tire size that fits those wood TOC Dunlop rims.
On and off, a company called Dean makes reproduction single tube tires costing multiple hundreds of dollars apiece. On and off, other makers supply solid display tires, and some can be ridden but they're heavy and sluggish. I have a friend who hunts down old-stock 700C "sew up" cyclo-cross tires and stretches them onto his antique rims. That seems to work, but most are much narrower than the original tires, and their gum sidewalls and offroad tread don't look ideal on an early bike.
If you're willing to forgo the original wood rims (there're likely not safe to ride on extensively anyway) you can relace the wheels with either 700C or "29er" touring rims or old-school 28 x 1 1/2" roadster rims. The latter is a size never used in the US but common in the rest of the world, and available from specialty suppliers on the internet. For the former, there was recently a rim made by Velocity with a profile very much like a wood rim. In fact, for a short while about ten years ago they even made them with a simulated wood-grain finish.
Good luck!