Not silly at all but you need training. Just holding a weapon is not enough, you must know how to react, how to use speed, how to fake, how to deploy, etc. I study FMA but focus on using a 3' stick as that is closer to a city cane. If you were in a rural area than 4'-6' might be of use.
Go with the cold steel route. There relatively cheap guaranteed not yo break made out a composite material and according to YouTube tests and demonstrations brutal
My great great grandfather, a Captain in the Confederate Army, made a cane from hickory and wrapped the handle with silver. It is thin and light, and even though it has gotten a bit warped from age, it is still strong enough to support my weight, and I'm not petite...
The silver wrapped handle would make a pretty decent war hammer in a clutch.