I'd be surprised if a very simple coinflip site, where there's always a 49% chance of the player winning, did not actually haul a 5% averaged profit or more due to gambling fallacies
This itself is a fallacy. It doesn't matter why people play, or what they're thinking. In a game with a 2% edge the house will average 2% profit on turnover in the long run(*). Even if every player played until they bust, that doesn't change anything. Some would get to play though their deposit more than 50 times over before busting, and some less. On average they'd play through 50 times and so lose 2% of their total wagered amount.
-So, then, why not offer a casino where the player has an edge per play (maybe even significant), but still has the cards stacked against them simply by gambling psychology?
Because that makes no sense at all. In a game of no skill, psychology plays no part. Actual returns approach expected returns over the long run. Offer a +EV no-skill game and quickly go bust.
(*) I know, Just-Dice has a 1% edge but only 0.33% profit. That's because "the long run" is longer than you think.