Taxing mineral extraction makes a lot of sense, because those minerals don't grow back (well, except on geological timescales). As for the income tax cut, well, I'd probably rather see it go to restoring the local government funding that was cut during the recession, but it's unlikely that Kasich or the GOP will go for that. They see local government budgets as bloated by far too many public employee union demands that were never seriously resisted.
Also remember that several states have no income tax at all.
I'm not following the logic on taxing it because its not renewable. Regardless, Kasich is 100% correct on raising the taxes because they are much lower in Ohio than they are in drilling meccas like Texas and the Dakotas.
Would I like to see him ease the pain of the local governments? Yeah, probably. But it's not like an income tax cut isn't going to ease the pain on small businesses (i.e. the job creators) and promote growth there. Simply put, it's not like he's taxing them and taking the cash and lighting it on fire; it is being put to a good use. Whether it's the best use is probably fair to debate.