They're much easier to turn and control in the oven for sure. I'm also the obsessive type that likes to constantly watch, look, and turn.
I was going to buy the same brush you linked but ended up getting this ash rake to scrape the stone instead: https://www.amazon.com/gp/product/B07JMR2R95/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1