Magnify NITRO™ 2

Puma Magnify Nitro 2, not the most fun midsole out there but it's a bouncy and sturdy shoe.

Puma magnify and magmax, Saucony Endorphin Azura (just came out), Novablast 5, Hoka Mach 6, (narrow foot friendly for sure), Tyr Maverick V1 are all lighter and protective but not mushy from my personal experience/research. The 1080v13 and v14 is also my go to casual/walking shoe and they increased the weight quite a bit from one to the other. But even then the 13 was a bit of a slog for me. I think modern trainers have the ability to pick up the pace and protect you on slower runs as well.

They're most likely just going to hurt - they're forcing your foot into a position that may not be natural for it. The most sympathetic support shoe to lower arches are the Mizuno Inspire, but they're dull to run in. Go try a few pair of shoes on and go with what's comfortable. You might be better in a more traditional drop (10mm); Brooks Ghost, Mizuno Rider, Puma Magnify are all great shoes.

I prefer the Puma Magnify over Magmax for long runs for the reasons you mentioned.

Don’t overlook PUMA 6-1 230 and use Magnify Nitro, really well built and not too soft. RIP to Salamon those things were tanks back in the day, they still make shoes but never found their niche on the road side and are all over the place.

I am 6'3, slightly lighter (82-85kg) and have raced in most shoe brands (<1:30 HM). My most important factors are (i) speed and (ii) comfort / toe protection. I train in Asics (Kayano - long slow runs) and Puma (Magnify Nitro - medium moderate run) and race in Puma (Deviate Nitro Elite 3). I am very happy happy with the pumas at the moment against my needs

Well if you want to go all out on your budget then Puma MagMax 2. It's wider than the Velocity Nitro 4 and can do your daily runs and longer runs and has the the grippy PumaGrip too. It will have everything you need for your half marathon training except speed sessions since while Nitro foam is bouncy, it is offset by its bulk. Edit: For badminton shoes you're going to need a different one. You need firm stable shoes for the sudden direction changes you're going to be doing during badminton and running shoes with their often soft foam isn't going to be good for your ankles