You’re basically describing options.
Well, with vanilla options, you'd need a basket of options to both reasonably closely replicate performance of a broad index (the "responsible vehicle" part) and also a small percentage of the time hitting a big pay-out. Presumably, you'd want mostly short-duration at-the-money calls on a low-fee broad-index ETF (so they often end up owning the ETF), and a small number of deep out-of-the money puts and calls on volatile micro-cap single names to get that sweet sweet irregular reinforcement dopamine hit.
This is not investment advice.
Maybe they way to go is lump-sum investing into some bond ETFs with gamified rebalancing into an age-appropriate mix of bond ETFs and broad equity index ETFs, following some pattern similar to value averaging for the rebalance, rather than cost averaging. It shouldn't actually be too hard to get something that has the right balance of feeling like real work, being relatively difficult to screw up (or the most common errors having little long-term effect), and out-performing dollar cost averaging.
This isn't investment advice, and even if it were, you shouldn't take investment advice from internet randos.
We need some sort of responsible investing vehicle that includes gamification, intermittent reinforcement, and a tiny chance for a huge payoff.