My MA advisor went to the top university in our field (think HYP..), and her fellowship was so good that she went on the job market never having taught a single section or course. She got interviews and offers everywhere she applied. Same goes for a professor who was hired into the history department at my SLAC undergrad - HYP, elite fellowships, zero teaching experience. You have to be approachable and able to give an excellent job talk, but prestige of program and fellowships seems to carry way more weight than teaching experience. I actually think having lots of teaching experience can work against you even and even be a red flag. At the R1 where I got my MA the students with the most teaching experience were usually the "weaker" ones, because they were the ones who couldn't land fellowships or external funding and were forced to accept TAships to fund their study.