Removing long-term residue from teflon

You cannot use physical methods (scratching), so you are only left with the chemical ones. And for this kind of gunk, the only thing I know to work is a strong enough base.

In the mildest cases, oven cleaner sprays might be sufficient. If not, somebody suggested ammonia in the comments, and then there is lye.

The big problem is that any base strong enough to clean this stuff will corrode aluminum too. If your pans are teflon on steel, go ahead and clean them. If they are aluminum pans, maybe you can try the oven cleaner as a last ditch effort, but you are risking the pan itself. Else, all you can do is continue use them as normal sticky pans.

The usual modus operandi is actually not to clean them somehow, but not to let the gunk happen. Just like you don't use metal on Termin, to prevent the failure mode of "scratched pan", you don't use techniques which produce gunk, to prevent the failure mode of "gummed up pan". This means that you have to avoid anything with small amounts of oil, and either use the pan without oil (at low temperatures though, or you'll burn the coating), or with lots of oil at moderate temperatures, or for wet cooking.

Providing this as an answer even though it's not really what I was looking for.

I didn't want to go the harsh chemical route - I ruined a previous version of the same pan that way, so I just got down with the old 'elbow grease'.

I went through about half a block of mica foam, a lot of Barkeepers Friend & another 'Astonish' cleaner with orange oil, and after about 40 minutes' work I got it back to a state where I could slide an omelette around in it with just a gentle shake, didn't even need to run a spatula round the edge.
I'd call that good enough.

Here's a suggestion I haven't tried (but might). If there are only a few scratches, protect them with a dam made of flour/water paste*, heated gently to dry it. If there's only one burnt-grease region, build your dam round that.

On the dirty side of the dam, apply any corrosive cleaner you like. Some, meant for ovens, are quite gel-like anyway. The scratched side can be left dry so you can see if the cleaner leaks, or can be wetted to dilute it.

Of course, you may have scratches and burnt regions together - but you might not, in my experience, burning in thin-based non-stick pans happens towards the edges (not much stirring, but a wide flame is almost underneath), while scratches happen nearer the middle (where more action takes place).

*Assuming you don't want to use Blu Tack, plasticine, or Play Doh on Cookware; we use them in work for making similar barriers (non-food).

According to this Reader's Digest 2/14/2020 Article, you should discard non-stick pans with significant scratches. The Teflon has been damaged and the chemicals can be flaking into your food. If there are scratches you are concerned that oven cleaner would get through, I would throw it away. Article Snipet Below.