Robins only eat bugs and worms as part of their raising an egg and feeding the chicks. Once the chicks can feed themselves they all switch to fruit and seeds.
That is absolutely not my experience here. My local robins eat bugs and worms all year, however, I do see them eating fruit (wild raspberries) here, when they are in season, and the fruit of some trees. I've never seen a robin eat a seed, only fruit. Robins also have multiple broods each year here. At least two, sometimes three rounds of chicks, so it's only the very end of summer that they aren't raising chicks (or having one following them around).
In many places the summer gets very dry often near end of year, and by then most of the insects are hunted out. That, along with fruit coming into season, may be one reason you're seeing this behaviour? I live beside a river and a wetland, though, so I have insects and worms all the time.
I wonder if we're talking about different robins. European ones aren't the same as North American ones, and I'm in Canada (currently in Quebec, but the same robins are in Ontario/etc too)
American robins come to my bird feeder quite often, but it's really for the peanuts (protein) and berries. The seeds themselves are deshelled, so they do occasionally take a sunflower chip. Shelless seeds have the side effect of attracting birds that don't normally visit feeders.
It's kind of cute: I'll see my resident robin observing the other local birds at the feeder and decide that he/she needs to get in on that too. They're smarter than they appear.
You're are correct though in that I've never need them take a seed that has a shell. I'm not sure their beaks are made to crack them open.
Birds like cardinals, chickadees, titmice and nuthatches do find insects for their young (protein), but primarily eat seeds the rest of the year. I'll still see them come to the feeder when they have babies, but it's for the high protein seeds like sunflower and also peanut pieces.
Funny enough, some birds, such as American goldfinches, don't feed their offspring insects at all to discourage brood parasites like cowbirds. Cowbird nestlings need insect protein to survive and if a cowbird lays eggs in a goldfinch nest, that bird is doomed.
If you really want to watch them up close, look for a feeder with a camera. If you're lucky you'll get some great video of them deciding what to eat and what to feed their young.
Mid Atlantic, I might be wrong about the seeds part. Their diet does shift in the winter as at 40 latitude they only migrate to closer to the shore and eat berries there until the spring.
