If you get a wood filler get an epoxy based wood filler. You may need to order it online, I’ve had trouble finding it at the normal hardware stores.
It’ll last forever and be more durable than the wood itself. Might be overkill for an indoor fix, but it’s not any pricier. I’ve used it to repair wood rot on outdoor window fixtures and it works great.
Just a word of advice. It’s tempting to add a bunch because you’re going to sand it down level anyway. But it takes FOREVER to sand if you don’t have a sander so try to make it level as best you can while you’re applying it.