Well "wood putty" covers a lot of ground, so a lot would depend on the exact product you meant. But given how deep the errant saw cut went I think you really should fill the gap with something structural, which would rule out most putties which are intended only for cosmetic work (and typically weak and crumbly).
I think ideal the way to fill this is using wood, and the grain should ideally be in the same orientation as the board. Although making such a thin cross-grain slice and getting it glued into the kerf in one piece will pose a challenge actually it doesn't need to be in one piece. Even if you had to resort to glueing in individual pieces cut from a narrower board once glued in place and the glue sets the pieces will act the same as a one-piece strip.
If you don't fancy the above idea I would resort to using thickened epoxy (epoxy1 + sanding dust2) as the easiest, very strong, alternative3. Place the board flat on the floor, tape off both ends of the kerf as dams and then, protip, heat your groove well with a hair drier or more carefully with a heat gun — when the epoxy touches the warmed wood it will instantly liquefy and flow into the kerf better. Now mix your batch of epoxy and spoon or dollop it in, then use coffee stirrers or other tools to help work it into the groove. While you can use any consistency you prefer I would recommend going no stiffer than approximately the consistency of mayo.
There are commercial epoxy putties sold for woodwork of course but their working times tend to be pretty limited, which along with the stiffer consistency will make filling a saw kerf bothersome. Plus they're not cheap, and as mentioned in the link included the 'Tootsie Roll' format they tend to come in means you never know in advance if what you buy will reliably blend to a smooth, homogenous mix.
1 You don't need a sophisticated epoxy, most dollar-store 5-min epoxy is actually pretty decent and good enough for this sort of thing. You might have to fill in 2-3 stages rather than all in one go.
2 If you don't have enough sanding dust available you can supplement it with other powders found at home, principally wheat flour or cornstarch.