Most trail cams are optimised for mammals at a meter or more. A key point to look out for is the activation time; the time between the camera detecting the beastie and it taking the picture. For birds visiting a feeder the bird may well have flown off again before the camera takes its shot. Another thing to look at is the close focus, most trail cameras are fixed focus and may not focus close up.
There are many different models on the market, I have experience with a few but never tried using one on a bird feeder. There are some products designed as dedicated feeder cams, often linked to an image recognition AI, that give you alerts to a phone if anything lands on the feeder. I have never tried such devices but they have mixed reviews, although this is mostly to do with the AI aspect.
As for counting things seen on cameras the standard rule is, it's your list count what you want. But most listing services where you compare lists with other people certainly wouldn't condone such things. The Ornithological Powers That Be do count remote camera captures, it might come to pass that a "first for Britian" is added to the list having been seen by no one.
A camera set up to capture images of Scottish wildcats in Sutherland has photographed a bird species rarely seen in Scotland.
www.bbc.co.uk
Bird Buddy is a smart bird feeder that notifies you of feathered visitors, captures their photos and organizes them in a beautiful collection!
mybirdbuddy.com