60/40 has advantages when trying to "fillet" a joint. SMD components are tiny and once the solder sets they stay put, however some components take a mechanical load and need support. The plastic phase of 60/40 lets you fillet the joint so it is a lot mechanically stronger.
I don’t mean to be rude, but this is literally nonsense. As in, it literally doesn’t even make sense.
What’s called the “fillet” in soldering is simply the shape of the meniscus caused by the surface tension of the molten solder. It applies equally to both leaded and lead-free solder — they produce identical fillets. If you’re getting radically different results when soldering with one vs the other, then something is radically wrong with your technique.
I don’t even know what you mean by “filleting” the joint — the fillet forms automatically from the correct application of solder — too little and you have a dinky, weak joint; too much and you have a convex glob instead of a nice concave fillet. (The component lead and PCB pad geometries of course are also defining factors in the shape and size of the fillet.)
Solder is NOT supposed to provide primary mechanical support. Of course, cheap stuff often relies on the strength of solder joints alone, but that’s bad design. (As in the picture you posted.) Especially with lead-free, which is very brittle, proper mechanical support is critical.
Here is a picture of the power connector from a relatively modern TV, a 29" Cello. As you can see the soldered joint has failed; the lead free solder is like water, and when it sets it is very thin and offers not a lot of mechanical support. I repaired it using 60/40 and got a nice filleted joint. Its been working for the last 2 years without problem.
Like water? What the heck are you talking about? As I said, the size of the joint depends on how much solder was applied.
Leaded solder is more flexible than lead-free, so it can handle flexing better without failing.
I sometimes turn down the iron temperature to make it more plastic to fill in loose joints.
Wait a sec — you mean you’re deliberately using a far-too-low temperature so you can glom on mushy-state half-melted solder to a joint?!?
I can’t even wrap my head around how bad that is as a technique. Solder joint strength relies on the fully-molten solder actually slightly dissolving the base metal of the component leads and PCB pads, so they fuse into a strong joint.