This past cycle I trained beltless high bar squat with Oly shoes. I hit openers yesterday for a meet coming up in a couple weeks. I tried belted low bar with Oly shoes and with flat shoes. The flat shoes felt a lot better even though I haven't trained in them in 20 months. With the flat shoes it feels easier to grind through a rep and keep everything tight. After the meet, I think I'll continue to compete in flat shoes while training in Oly shoes, at least for the next year. My posterior chain still feels significantly stronger than my anterior chain even though I've focused less on it this past year. There's nothing wrong with using a variation over a comp lift if you want to bring up a weakness or find the movement more effective for building strength. The comp lift is for displaying that strength.
High bar squats really helped to strengthen my quads, glutes, abs and upper back. I put a lot of focus on keeping my hip flexors tight as well since there's less emphasis on them with the more upright torso angle. Doing high bar while still doing lower back and hamstring work makes it easy to switch back to low bar. I do light work with good mornings for one session, medium work with RDLs for one session and heavy work with sumo rack pulls for another session. My deadlift feels stronger as well. I feel like I get more carryover to low bar squat and sumo deadlift from high bar squat compared to front squat. When running the cycle with front squats, I focused too much on my glutes without getting my hip flexors tight so that might explain why.
Even though you want to focus on your quads, I would still recommend to generate as much tension in your hips as possible. When you keep your hips stable, your knees remain stable and this allows your quads and glutes to consistently do the same work.