No if you are walking, then EIS or a gimbal are a must have as otherwise the footage get very frustrating bumpy footage to look at.
If you only have a phone to view or share footage from that is a limitation too worth considering, i automatic assume everyone have a PC as that's the kind of guy i am.
You can also playback on most modern televisions as they have USB ports you can plug a card reader into, and a build in player,,,,, i personally have too old eyes and a much too much hate for phones to look at phone screens, I only have a phone so my dear old sick mother can get in contact with me.
The wifi transfer speeds from the wifi of any camera are still quite modest, but once you have gotten footage onto your phone as i understand it they or at least your P30 have ample power to edit the footage there.
So moving masses of footage from a camera are still pretty slow, but your only option as you just have the phone, in general it is much faster than the speeds you will get off the memory card itself in a good USB 3.0 card reader ( full red speed in a card reader so card read speed determine that and that's often around 100 MB a second or up there at least, where wifi can be down around 20 MB second transfer speeds )
The new action cameras like the new gopro and dji osmo action and so on have almost unreal electronic image stabilization, so you can litterally jump around on one leg with the camera in hand, and it will look as if the camera are on a gimbal or hovering on a drone, the F1 do not have that kind of image stabilization, but what it have should smooth out regular walking at least.
Video with EIS working and violent shaking cameras.
Runcam and foxxer have also made cameras in the same shape category as the F1 and mobius, but these like the F1 are also on "old" hardware, so on them EIS are also not as super as on the very new cameras.
You might also consider a 360 camera like the insta 360 one x, but that will not mount on the brim of a baseball cap, but it do have very good EIS and you have a 360 degree footage in which you can pick out a regular flat footage to export.
The good thing here is improved EIS ( Electronic Image Stabilization ) over the F1 for instance.
Most new action cameras support headstrap or a chest harness, but even though they are wide angle, you still have to point them in the general direction of what you like to capture, so for a dog scouring around at your feet a head strap might be the better of those 2 options,,,,,, though it do look a bit silly with a "headband" like that and a camera on your forehead, bit the camera will always be pointing in the direction your head are facing.
Newer action cameras like the osmo action ASO are however a bit more expensive than a F1 for instance.