What an awesome thread!
I agree with points on both sides of this issue but being a mechanic will bring tools and suspect spares with me. The fuel filters are a must, period, must have IMO. Other things, well, you decide
I have been wondering about the spare tire thing also so I appreciate the heads up on the Roadmaster tire carrier. A bit pricey but having tires that are very hard to get, can get you going much faster.
I don't go out of my way to buy spare items just to carry with me but I do see the value in knowing what items you might expect to need so great question. Agreed though, if you have it, you may not need it.
I would likely throw some engine coolant, maybe some radiator stop leak (just for an absolute emergency), generator belt, generator filters, spray lubricant, a mat, rug, or something "thin" than you can lay on under the coach in case you have to go under it and can't raise it much, maybe a heavy duty jack, good flash lights, fuses, solder, solder iron, wiring diagrams, owners manuals, a few misc pieces of hardwood, and so on
All that assuming you can do anything with those items if you need to
I hope that you have an amazing trip!