Frozen Solid
An eavestrough full of ice is not a drainage fault. It is a heat-loss fault, and the ice is the symptom.
Short answer
Heat escaping into the attic melts snow on the roof, the meltwater runs to the cold eave and freezes in the trough, and each cycle adds to it. A trough that is blocked with leaves holds the water that then freezes, which makes it worse. The fix is in the attic and in the fall cleaning, not in chipping ice out in January.
Why it freezes
Warm air leaking into the attic melts snow on the roof above the heated space.
That meltwater runs down to the eave, which overhangs unheated space and sits at outdoor temperature.
It freezes there, in the trough and at the top of the downspout.
Each melt cycle adds another layer, which is why the problem builds through January rather than appearing in one storm.
A trough with leaves in it holds standing water, which freezes first and then dams everything behind it.
And once the trough is full, the ice builds up onto the roof edge, which is an ice dam.
What the ice weighs
A long run of eavestrough packed with ice weighs enough to pull the hangers out of the fascia, bend the trough and in some cases tear the whole run off. That is the usual damage, and it happens in February rather than during the thaw.
The damage it causes
Hangers pulled out of the fascia, and the trough sagging or detaching.
The trough deformed permanently, which then does not drain even when it thaws.
Downspouts split by ice expansion, usually at the top elbow.
Water backing up over the roof edge and under the shingles.
Fascia and soffit rot from the water running behind rather than into the trough.
And a wet foundation in spring, because the meltwater went straight down the wall instead of through the downspout.
The fix, in order
Clear the trough and the downspouts in the fall, after leaf fall. This is the single most effective thing and it costs an afternoon.
Air seal the attic ceiling plane: pot lights, top plates, the hatch, plumbing and wiring penetrations.
Duct the bathroom fans outside rather than into the attic.
Clear the soffit intake and fit baffles.
Insulate properly, with the eave area done rather than skipped.
And only then consider heat cable in the trough and the downspout, which manages the symptom and is a legitimate measure on a stubborn north-facing run.
Common questions
Why does my eavestrough freeze every winter?
Heat is escaping into the attic and melting the snow above. The ice is the symptom.
Should I chip the ice out?
No. You will damage the trough. Clear it in the fall and fix the attic.
Does heat cable help?
It manages the symptom. Useful on a stubborn run, after the attic work rather than instead of it.