I guess that it depends on what type or riding I am planning on doing. For a nice leisurely ride, a hybrid that lets me sit up is good. Also easier on the neck and shoulders. For a longer or harder ride, I prefer my Trek Crossrip. It’s a sub-cyclocross bike that really handles the trail and gravel roads very well.
This will be a judgement call on your part. Overall, the trail is pretty firm. It is also smoother than a gravel roads because the chat is much smaller than they tend to use on roads these days. (Seriously Kansas, is the budget so bad that you have to throw medium sized rocks onto your roads?) I have talked to many people who have made the trek with a road bike and had no problems.
However, often after rains you will find soft spots where a road bike would sink in a bit. I’m not talking quicksand here but you will definately get a tug on your wheel. If you are alert this is no problem, but a surprise can lead to a comedy routine of overcompensation followed by tragedy. YMMV.
@J-rider, road bikes typically do fine. I ride a bike with 28mm tires on it often, and that’s no problem. My friend rides the trail with 23mm tires, and it works.
As Timon said, the trail can be soft after rain (or snow), and narrow tires tend to cut into the surface, not only making it hard to maintain momentum, but creating ruts that will eventually dry out and create a bumpy surface. So that situation should be avoided if possible.
I have an old Cannondale mtn bike frame completly rebuilt with street slick tires. Really like it for this type of trail. Put a 27 sp. drive train and long bar ends to change riding positions. Very good on hard pack limestone even when wet. It is al.frame but no front shock so is pretty light. Have riddin KT trail, Prairie Spirit and took on Amtrak to New Mexico. I like it because it is not a theft hazard and is easy to take on bus or train or throw in back of pickup.