You can visually inspect the brushes, not that big of a deal. Mark the three starter sections to make realignment and reassembly easier, just in case a section gets twisted so you you don't have to go fishing with the long bolts. Take the back end cap off.
The braided wire going to the brushes is pretty stiff, as the brushes get shorter, the spring pushing on the end of the brushes gets weaker. You can sometimes rearrange the wire a little and get some more use out of the brushes.
Tapping on the side of the starter, can seat the brushes in the holder a little deeper and get you a few more starts.
Jumping the starter at the solenoid may help the brushes arch the gap, the wire from the starter relay to the solenoid is pretty anemic and can choke off some of the current.
Getting the end cap back on the starter is a pain, but doable and is a lot easier with three or four hands. You have to compress the four brushes to get the cap back on.
Taking the solenoid off the starter is to be avoided, the bolts are almost guaranteed to be be welded in place and there is a good chance of snapping them off. Solenoids rarely fail anyway, I've only seen one bad on in 26 years and that was rust from a torn seal and not an electrical failure.
Mud in the brushes (yes it does find it's way in there) and oil on brushes (yes it does find it's way in there) does about the same thing as worn brushes, you get iffy function on the starter. Sometimes the starter works and sometimes it doesn't, it all depends on enough current finding a path through the brushes.
It may be something else, but brushes, especially on a thirteen year old starter, are a prime suspect. It has been my experience the brushes last around 6 years depending on how often the starter gets full of muddy water or oil. It the starter gets full of muddy water (and/or sand) the brushes can wear down pretty quickly.
A new brush is 3/4" long, when it wears 3/8" the wire going to the brush jams against the brush holder.