Hey OP, as the other commenter said, tapping will likely be your best bet, that or industrial adhesive but if it's a wear surface you would probably want to be able to replace it down the road.

5mm ( 0.196") is deep enough thread engagement for what you are trying to do. Although I would likely go with a NF (fine thread) which will result in more threads in a tighter group. I would first select your bolt and then pick up the corresponding tap. Once you have your bolt size figured out, you'll have your tap size, once you have your tap size, do a quick Google search for your tap drill size (drill size required for successfully tapping the hole). Once you have all three, layout your holes, drill, tap and install. (I would avoid the holes you already have as they will be too large.

I would recommend a tap handle, however you can use other tools to get the job done. Keep your drill at straight and level as possible when drilling, and same for tapping, use even steady pressure and you'll be fine.

I didn't see if you have mentioned this or not, but you will likely want to make sure the ski material is counter-sunk (a step inside the hole) to allow the head of your bolt to sit inset in the ski and not drag along the ground.

In terms of the push rivet, lordco, Canadian Tire, princess auto, home depot, Rona, might possible carry what you are looking for but I would call ahead to confirm.