I haven't tried to carry on my bike on Amtrak, only used the trainside baggage service. On the Sunset limited trainside checked baggage only works at stations with baggage service. For example, the Sunset limited stops in Deming, Lordsburg and Benson (AZ). However none of these are baggage stops. The nearest are El Paso TX and Tucson AZ. I used both in 2019 on a trip where I took Amtrak from Austin to Tucson, cycled to El Paso and took Amtrak back from El Paso. So if you want to take your bike on board then I would follow advice of others as far as having a folding bike, etc.

If instead, you want to bring the bike in checked baggage service, then the tires seem to me to be within specs. In any case, the ticket offices open somewhat in advance of the train departure and to use the trainside service I have always reserved a bike slot at the same time I book my ticket. So I show up around the time the checkin happens since (a) I want to check a duffel bag of luggage with my panniers and (b) I need a baggage tag on my bike to hand it to folks doing trainside luggage. I doubt this happens but if there were somehow an issue, the conductor would tell you at time you check in - so I would bring tools to take off pedals and turn handlebars and lower seat just in case. A little later when the train is about to arrive and everyone lines up for boarding, I go out with my bike and most all the time, they already direct me to go to baggage car. I hand my bike to someone in the baggage car. There are hooks in the car, though they don't always hang the bike by the hooks and sometimes just put the bike in the baggage car.