It's more likely that your iron is under powered for your application, and frankly a copper tip will likely make the problem even worse. There are three dimensions that impact heat transfer (temperature being equal): Thermal mass of the tip, thermal conductivity of the tip, and power transfer to the tip.
If your iron cannot support a regular tip, it's likely that it cannot get enough power to the tip to sustain the heat being pulled from it (copper will just expedite this). You can try a beefier tip (or load up a tip with wet solder which can kinda give the same effect) or you can get a higher power iron.
I've been soldering for 15 years with all manner of irons, I've never needed a copper tip, but I often need more than your standard hakko or weller. 400C on a the hakko is not at all close to being equal to 400C on a Metcal for instance.
When the tip is tinned effectively though, my Hakko works fine. The problem is keeping it properly tinned is a damn challenge: i.e. I think I wiped the whole thing on a sal ammoniac block, then rosin flux, then put the whole tip into a solder bath and got...a tiny spot on the tip which has remained reliably tinned and is the one spot which works well.
The symptoms are pretty obvious: even if the iron was underpowered, melting a thin piece of solder wire should be pretty much instant. My issue has just always been getting the tip to a shiny solder finish with all the recommended tools...just doesn't work once that iron layer oxidizes initially.
EDIT: Although thinking about this now, actually the way you get rid of mill-scale is with white vinegar so maybe that's my actual answer: white vinegar for a couple hours, then rosin flux and then into the solder bath. Hopefully won't remove the entire iron coating on the tip since otherwise...copper tips again.
I had a tip like this once. I ended up getting an old flat blade screwdriver and running the sharpish edge of it up and down the tip to remove the baked on crud, until I could see the dull shine of the plating all around. I then tinned the tip using flux bearing (electronics) solder and repeated until all the dry bits on the tip had gone and the tip was fully tinned. It seems to have been okay since then (quite a number of years).
I wouldn't want to do the above too often and it's probably not the recommended way, but I seem to have gotten away with it by not being too rough with the screwdriver. Potentially worth a try if the tip is a write off otherwise.
Once it is tinned, clean and tin the tip before turning off after every use (but you probably already know that).
It sounds like maybe either you're doing something wrong or there's something wrong with your iron. I have a Hakko and it works great, I have no issues with tinning the tip at all, never had to do a vinegar soak or anything like that.
If your iron cannot support a regular tip, it's likely that it cannot get enough power to the tip to sustain the heat being pulled from it (copper will just expedite this). You can try a beefier tip (or load up a tip with wet solder which can kinda give the same effect) or you can get a higher power iron.
I've been soldering for 15 years with all manner of irons, I've never needed a copper tip, but I often need more than your standard hakko or weller. 400C on a the hakko is not at all close to being equal to 400C on a Metcal for instance.