I wouldn't think such a quick "seasoning" on the stove top would be enough after removing all that old crud. What you're really going for is a reduction of the oil to a thick tar that then solidifies on the pan and seals it.

(I put seasoning in quotes because I really hate that it's called that, since it's really just sealing the pan.)

It's probably good enough to use, but I like to do a longer "seasoning" in the oven with multiple wipe downs and re-oils over a period of about two hours. I do this either when reconditioning, or after a pan has had the seasoning damaged due to burning or just cooking something that for some reason makes it stick a bit more. Acidic foods, particularly with a lot of water can do this.

I sometimes do a quick stove top seasoning to build up the layers of seasoning, but I don't feel like it's as easy to really bake on the layers of oil to seal in the cast iron, protect it, and make it more non-stick.

Of course, I still use my iron skillets plenty even when I think I probably need to go through another seasoning of them, because I don't want to get all that oil in the air when I can't open the windows. So it's not like they're not ever unusable, even when they don't have a particularly thick layer of seasoning to make them non-stick. They just cook better the thicker the layers.

If you just make sure to cook mainly oil things in it for a while, a stove top seasoning is probably fine. Just might not be the most non-stick pan in your kitchen until it builds up a bit more.

When I recondition one though, I usually clean it first with soap and vinegar to get all the crud off, before hitting it with a wire wheel, then cleaning it again. The first clean is mainly just to get the stuff off so it doesn't go all over the place when I hit it with the wire wheel. The vinegar also removes rust. I usually hit the whole thing including the bottom and handle with the wire wheel, so I can reseal the entire thing when seasoning.

Seasoning the bottom isn't entirely necessary really, and much of it will burn off when using the pan on the stove, but it prevents rusting and damage. It also burns up a bit when you're first using it though, which doesn't smell the greatest. Some oil builds up over time on the handle, sides, and bottom over time with use as well.

You made me wanna re-season my cast iron. Sadly I can't do it today, because I can't open the windows.