Remember fresh gas, good spark, good compression, & fresh air at air filter are needed to start engine. I know Manual says valve adjustment is .004 to .008 & you set both at .005, why not set both in middle at .006. I always use high test gas & Star Tron fuel treatment or B&S fuel treatment. Also try a different multi-grade oil instead of straight weight. 10w-30 or 5w-30 I use Full Synthetic 5w-30 in my B&S 20 h.p. Vanguard & change filter every time, either Valvoline or Pennzoil because they've been around more than a century. Walmart is where I purchase oil, has best prices.
Auto parts stores loan tools out with a deposit which you receive back when you return tool, so no cost to you.