1. The fact that there are threads asking for hyphenation support in Designer dating back to 2015/2016 is as good as a "no". Besides, literally after enabling the option in Publisher and switching back to Designer, it stays active and working perfectly. That means it IS already implemented in Designer, they just hid the tickbox to enable it, as a form of primitive "DRM" to see if you purchased Publisher. It's only a matter of time before some hacky solution comes along to enable it without Publisher, but it might require shady addons from shady people on shady sites, putting Affinity users in danger of malware attacks just because Serif refused to add the damn tickbox. This is not okay.
2. I'll trust what you say about Publisher is true, since I've never used the program, I just know it has a Designer persona. It might be an awesome piece of software, and THAT SHOULD be the reason people buy it, not to enable features in other programs that require it as a proof-of-purchase or somesuch.
3. Why buy and work in Designer in the first place then if you can do everything you can do there in Publisher instead? Truth is, the programs have different goals; it's true that Publisher is more catered towards text manipulation, but that doesn't mean Designer can't have a simple feature such as Hyphenation, given the code is already in the program! My work is better suited to Designer, since I need to use complex vector graphics and effects in addition to text manipulation and I'd like to be able to do it one program, not 3. This is the reason I purchased Designer, because it implemented raster editing in addition to vector instead of having to rely on yet another program, and then they go and do things like this. It goes against their main strength and ideology.