If the newel post is next to the stair then I definitely would lag into the stair as well as from underneath. If you’re relying solely on an underneath connection then a lag bolt is your best friend but the lag bolt has to be through more than just the subfloor and hardwood. You gotta have some sort of 2x or 4x material underneath the subfloor that is securely fastened to whatever then lag through everything. When shit hits the fan and somebody grabs onto the newel post because they’re falling, it has to be securely fastened. This is the worst case scenario but that’s what you have to plan for otherwise why not just hot glue a plastic post in place for looks.
When I’m building things like bunk beds n such in trailers I build them as if I were the one who would be using it. People, fat people, are gonna be drinkin and doing who knows what out in the woods, that’s what I have to build to. Your newel post is no different. Yeah it’s gotta look nice but it’s real purpose is what happens when some fat F falls and needs to grab onto it for support. That’s what you have to build to. That’s what it’s purpose is. Think of a newel post as a structural part of your house. Or maybe you can think of Aunt Ethel falling down the stairs cus the newel post gave way when she fell. Now you’re gonna be the talk of Christmas until the end of time.
Since you have access to underneath I would definitely mount the newel post as Havasu has described. Cut the hole tight. Maybe you use a multi tool as opposed to a skill saw, doesn’t matter how you cut the hole, just cut the hole tight. Now you’ve sunk the post through the subfloor, and your hardwoods, now you need something solid to fasten it to. Generally there is some framing down below that you can lag into, maybe you add some. Doesn’t matter how you do it as much as the fact that you did it.
As far as lagging into the stair from the side, which I’m a fan of, you can always bore a hole to countersink the lag then plug the hole.