Why is my fridge water dispenser putting out little white flakes?

Whitish flakes after a filter swap on hard water is usally one of two things: calcium scale that got knocked loose, or gunk growing on the old filter that ended up downstream of it. A new filter changes the flow and pressure through the line a bit, and that's often enough to break loose stuff that had been sitting in the tubing for years. The fact that your ice is fine actually points away from the fridge internals degrading. The ice maker and dispenser typically share the same supply, so if something inside the fridge was falling apart you'd expect to see it in both.

Here's how I'd narrow it down. Pull the filter and put the bypass plug in, then run the dispenser and let a glass sit still for a few minutes. If the flakes stop, the filter itself is shedding something, either carbon fines or material from the housing, and you'd want to try a diferent filter. If the flakes keep coming with no filter in the path, it's scale or buildup in the tubing and the filter just stirred it up. Also take a look at the dispenser nozzle itself. That little outlet area gets surprisingly gross, and people almost never clean it. Mineral crust and slime build up right at the spout and flake off into your glass, which would explain why the ice looks normal. A old toothbrush with some vinegar on the nozzle and the recess around it does alot.

Since your water is hard, chalky taste plus white flakes that might settle in the bottom of a glass points to calcium carbonate. You can test it yourself. Sprinkle a few flakes in a bit of warm vinegar. If they fizz or dissolve over a few minutes, it's mineral scale and totally harmless, just annoying. If they just sit there or get slimy, it's more likely biofilm, which means a longer flush and maybe sanitizing the line is in order.

I'd keep flushing. Gallons, not glasses, and fill a pitcher so you're not standing there. If it hasn't cleared up after a solid flush with the bypass test done, then I'd be looking at the filter brand next.

Did appliance service calls for a few years and the thing people miss is the internal reservoir tank. On most fridges that's a coil of tubing sitting back behind the crisper drawers holding a couple liters of water at all times, and it never fully drains when you swap a filter. That's why the first few gallons after a change still taste off or look weird. Letting a glass sit for ten minutes also tells you something, real scale settles to the bottom fast while plastic or rubber bits float and stick to the sides.

Went thru almost the same thing a couple winters back and the culprit ended up being upstream of the fridge, not the fridge itself. The shut-off valve under the sink had some mineral crud in it and every time the dispenser pulled water it sent little bits down the line. What helped me was keeping a plain white measuring cup by the dispenser, the flakes are way easier to see against white and I could acutally track wether flushing was working. Mine cleared up after about a week of occasional flushing. If yours drags on past that I'd peek at that valve and the supply line before blaming the new filter.

New here so hope this is relevant, but I had something similar in my apartment fridge a while back and it turned out the filter I bought was a cheap off-brand knockoff of the one the fridge was made for. It fit fine but the water was cloudy with little flecks untill I swapped it for one that actually matched the model. Might not be your issue since yours started right after the change, but worth checking that the replacement is the right one and not a lookalike. I'm still not sure how anyone tells them apart, the boxes look nearly identical.

Not sold on the ice logic in the first reply. Sure, the ice maker and dispenser pull from the same line, but the dispenser blasts water through fast and the ice maker fills at a slow trickle. Loose bits in the tubing are way more likely to get kicked up and carried out by the dispenser flow, while the slow fill never stirs them up. So clear ice doesn't relly rule out anything inside the fridge. Try running the dispenser at a trickle into a glass instead of full flow. If the flakes mostly show up at full speed, that says it's stuff getting stirred loose, not the filter shedding.

Update: pulled the filter out and checked, it's the actual brand for the fridge so probly not the knockoff thing the new guy mentioned. The reply about that coiled reservoir tank behind the crisper drawers got me thinking though. Flushed several more gallons and the flakes are fewer but still showing up. Would that tank explain why the ice stays clean since it draws from a different path? And is flushing from the shut-off valve under the sink worth trying next?