I would strongly recommend purchasing a rubber ferrule from one of the many stick making supply businesses that sell bits and pieces for stick making. I have a very simple hazel stick and started off with a metal ferrule (not a spike) and found that it slipped and skidded on rocks. The rubber one was a simple push on and has been solid for many years now. No need for mud baskets, unless you plan to use on snow, in which case...errrr...no idea. No need to "reinforce" the lower end of the stick, beyond fitting the rubber foot.
If you do not fit a foot, and only walk on soft ground, the stick could still hold up for a a very long time. My mum has a broad leaf maple stick that I made over 30 years ago, no ferrule, no foot, just the wood. She only uses it in woodland, or on the sand dunes at Merthyr Mawr. If used on roads, or rocky mountain trails, rubber is the way to go for a home made stick.
As for a DIY ferrule, well, the bought ones are very inexpensive and matching the durability in a DIY solution is going to be difficult I think. Certainly more time consuming than the job warrants. Next best I suppose would be some sort of tube, maybe small plumbing or copper tube, with rubber bungs or sealant up inside it, slid on and glued to the end of the stick.
I am interested to know how the birch turns out. Most of the young birch in the UK that I have seen have a little too much taper in the length for what I would want in a stick. Local conditions probably make a lot of difference!
Good luck
Chris