agree with gene... when you kick the starter over it pulls a TON of AMPS to spin the starter !! what you are describing really sounds like a battery on its last leg ? been there recently. does it spin it at full speed like normal or just slowly ?
1-would check cables, mainly ground (make sure it is robust and connects to clean bare metal on the frame) and that the power that goes from battery direct to the starter is a robust cable and no corrosion on either end. i use 00 cables for power and ground on my 1929.
->you can check if it is the ground cable by using a set of jumper cables and connecting the ground from battery negative to one of the bolts holding the starter to the housing, or clean metal close by on the trans, engine, housing.
2-battery (how old is it ? typical life is 3-5 years (can last longer +2 years if battery spends life on a maintainer when not in use) can test this as well if you have a spare 6v around, if not you can use a 12v in parallel with the 6v, but do NOT leave it connected !! i put the 12v on the ground near the battery, run jumper cables from 12v to 6v in parallel (+ to + & - to -), i do NOT connect the ground until i am ready to try and start the car. do this all by hand, i make sure all electronics are OFF (lights, etc) then turn on key, and push the starter pedal with my left hand as i touch the 12v negative to the 6v negative. see if starter picks up and spins harder and faster. 12v will NOT hurt the starter only electronics in the car (lights, etc) only do it for about 5-6 seconds then pull away the 12v negative and release the pedal.
the starter switch is just a simple contact switch. can look it over and see if it is gunked up, but they typically last forever. remove power and then the 2 screws and they come right off and you can see the simplicity. if bad they are readily available on the market NOS, NORS, etc...
you can also pull the starter and put in on your bench, i put mine in my vise and used jumper cables to connect to my 6vbattery to make sure it would spin up. if it is bad complete rebuild kits rare readily available. have rebuilt 2 of them to date. complete rebuild kit has armature, brushes, bearing/ bushing, and bendix.