I have seen a bird shot taken with a Panasonic 1 inch sensor, that would make any birder proud. However, I've only seen one.When comparing two camera systems there will always be circumstances where one better than the other.
The circumstance you describe is probably one of them.
The smaller sensor will have more depth of field at the faster ƒ-stop. ƒ4 on a K-1 or even K-70 the DoF is going to be quite narrow. AT ƒ4 the 1 inch sensor will get much more of the image in focus by extending the depth of field.
I take advantage of this with my XG-1 shooting macros on the forest floor under the dim light of the canopy.
XG-1, every thing in focus at ƒ5.8
K-1 @ ƒ5.6 almost nothing in focus....
I like both images, but neither could have been taken with the other camera.
I like having both available, for when i want an image the other won't do.
I can stop the K-1 down to ƒ16 or ƒ22 to approximate the XG-1 image, though that will take the 2sec. timer and a tripod and a long exposure (and that technique absolutely will not work on a stage production, great on mushrooms though.). There's nothing I know of you can use to get the K-1 image with the XG-1.
The fact that you can do more with the larger sensor, in no way implies their aren't times when the smaller sensor won't have the upper hand.
There is a sort of internet myth that getting a larger sensor gets you better images in every circumstance. There will be times when the advantage goes to the smaller sensor for a of a host of reasons, one of them being, what you want the image to look like. The nonsense that the smaller sensor is just a crop of the larger is inaccurate on many levels, including DoF for given a given aperture. If you have to shoot in low light and need depth of field, a one inch sensor could be the best choice, but it will be based on many factors. ISO, Aperture/shutter speed ratio, there's definitely a 1 inch sensor sweet spot there. Maybe you and your friend found it.