I don’t use fleece because it collects “stuff” which irritates the skin. Seeds, chaff from hay, bits of wood, splintered pieces of shaving bedding.
Wrapping parts of the halter using fabric is easy. Cut the fabric big enough time be the length of the strap and wrap around a couple times. Then secure it on the strap with a couple pieces of tape. I use athletic wrist and ankle tape. Cheaper, better glue than electric or adhesive tapes.
Horses here don’t wear halters a lot, but may get “greasy” places on halters that irritate summer hair or skin, so halters get wrapped temporarily. Fabric used for wrapping is considered “disposable” so I toss it when removed. Never used flannel, consider it too thin unless wrapped a number of times. I prefer old towel pieces, washrags are a perfect noseband size for my horses halters. I want any wrapping to also pad the halter strap so weight is not rubbing skin.
We use nylon halters, so they are washed often, not allowed to get crusty dirt buildup. I wash them in my washer (top loader), in net laundry bags, hang to dry. Might take two cycles t get clean.
Leather halters would have to be hand scrubbed with a brush to get the grunge off, clean the stitching. We never use leather halters because they require more work to keep clean, leather supple and they break under strain.
I would not use leg wraps as halter coverings unless cut into pieces. Polar fleece picks up as much trash as fuzzy (sheepskin, real or fake) fleece, so not my choice as halter padding.
ADDED; I posted before your picture. I would check around for a halter with the strap ends on the outside, not against the shin. Looks like that halter is sewn with the strap ends tucked INSIDE so sealed nylon and stitching is against the skin to rub. Not made by horse folks!!
I have never had rubs in those locations, since my Valhoma halters are not sewn that way. Mine are not breakable which some stables require when they also require halters to be left on during turnout.