Diablo 3 is fundamentally broken

[Lissa]

can only craft (up to) ilvl 62 weapons. (up to) ilvl63 are only drops.

You can only craft ilvl 62 weapons if you get an ultra-rare pattern drop to teach your blacksmith. Which may or may not be the base item type you're after. I think crafted (high level) weapons are also limited to 4 affixes, while drops can be 6.

In general, mlvl 60 monsters are Act IV hell, 61 Act I inferno, 62 Act II, and 63 Act III/IV. Similar to Diablo II, just having monsters of the right level doesn't guarantee a high base item type, and there is only a small chance to even have the right base item type (and level) for the drops.

I think the crux of the gear issue is that they streamlined too much.
Every class wants DPS weapons. So once you're level 60, everyone is looking for the same 2 affixes types (+% damage and +xxx-xxx damage). Plus whatever their class flavor affixes are.
Similarly, every class needs a baseline level of +all resist, so armor without a +all resist on it is an order of magnitude less valuable.

In D2, the affixes were more spread out. There were builds that wanted +skills and low damage was okay. There were builds that wanted high damage, and no +skills were okay (but additional skills helped). This meant the community as a whole could share a little more. Also, +all resists was a unique only attribute, so you had to mix and match individual resists. This was kind of a pain (especially since there was a resist cap) but it meant that more gear was useful to more people. It seems to me, there's just too high a percentage of people after the same affixes due to the game design.

No, there are patterns for crafting 6 affixes for both weapons and armor. The patterns are as followed:

Exalted <item> - 4 Affixes
Fine Exalted <item> - 5 Affixes
Grand Exalted <item> - 6 Affixes