Humans have been running barefoot for thousands of yearrs. While shoes are only a relatively modern invention.
And one of the downsides of shoes is that it leads to a lot of the muscles in the foot that normally provide shock absorbtion, atrophying and getting weaker.
Even the staunchest advocates of minimalist/barefoot running suggest switching over only gradually, starting with shorter runs and softer surfaces only, so as to give your foot muscles time to get stronger.
Particularly as it sounds like your foot muscles are pretty weak, I'd recommend going with a more conventional type of shoe (and something that suits them better than your current pair) while training for a half marathon, and only converting over to something more minimalist later.
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.
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