Echoing
@Gkprevite here... I have been running with a
SPIbelt for years without issue, although having the keys and phone in the same "pocket" may risk scratching the phone. I think I have had mine for about 5 years at this point and the elastic is only just starting to show signs of wear.
I've found that most shorts have some sort of very small zip pocket
somewhere. I am a big fan of the
Saucony Split shorts, if I need to run with a key. The Saucony shorts have a weird butt pocket but you don't notice any bounce that you might with a side pocket. I also have a pair of New Balance shorts (honestly I am not sure which style exactly) but they have a surprisingly large side pocket - so NB may be worth looking into if you're really married to the idea of a side pocket big enough for a phone.
I really hate running with stuff and I used to just hide my key somewhere nearby. I realize that is not practical (or wise) in all situations. Because my hatred of having stuff in my pockets extends beyond running, I also recently installed a smart lock so I can just open my door with my phone. So long keys!