How can I fix gaps and broken mortar in a stone path
It seems that your problem is likely due to movement of the supporting soil under the stones, so your idea of a bordering strip to stabilize lateral movement is a good one. A strip of concrete only an inch wide would be difficult to create and probably ineffective. You would be better off with a larger border.

The method you choose to create a border would depend on how you want it to look, how much work you are willing to put into it, and what materials you are comfortable working with. Concrete would work if formed and poured properly. I would probably use a quick and easy method, like pounding rebar in to support partially buried pressure-treated dimensional lumber (1x4, 1x6, 2x6, etc.)

You will need to pull up the stones in the whole area, create your borders, than backfill with sand and tamp well until the area is solid and level. Then replace the stones and fill the joints with sand and tamp again.

Kinda like the picture below, except with lumber instead of flex border: