If I equip Flint to a Spirit Reaper sucessfully, will I be able to use Flint's reattachment effect on another monster?

Here's the text for Flint . . .

Since the text does not specify that the monster needs to be destroyed in a certain manner, it should work with Spirit Reaper in the manner you are asking.

As soon as Flint's equipping resolves, Spirit Reaper will now trigger and attempt to resolve it's effect. Should it be destroyed Flint will now trigger and attempt to target another monster for the effect.