You could implement the method of opting into a cumulative ranking leaderboard in a similar way that opting into the leaderboard of any other game mode is done.
Before I begin, I would like to mention that I'm going to call this proposed mode osu!omnithlon, combining the prefix omni- and the suffix -thlon. Omnithlon is chosen over quadrathlon to support future game modes.
The current way to get counted into the leaderboard of any game mode is to just play enough of that game mode. The same can hold true of osu!omnithlon. This can be done by either changing the game mode to omnithlon, and only players that want to be part of this cumulative would have a ranking for this mode displayed on their profile.
Thinking about this mode further, I've come across the following realizations. Walls of text lay ahead.
1. pp between game modes is not equivalent.
If we take a glance at the top player of osu!standard, his top play as of writing this post is worth 817pp. Currently, scores exceeding 800pp are extremely rare and, as far as I'm aware, the only other play that rewards at least 800pp is held by the same player.
If we look at ctb, finding plays that exceed 800pp are not hard to find amongst the players at the top of the leaderboard.
Because of this, osu!omnithlon, if implemented, should have some kind of pp equivalency rate, where pp rewards are somehow made roughly equivalent based on skill level. This can be done by finding an average of pp score ratios between modes, but there will without a doubt be countless debates on what amount of skill in one game mode is equivalent to what amount of skill in another game mode. With the implementation of this kind of system, this debate is unavoidable. Not balancing pp would lead to the omnithlon leaderboards being dominated by players of just one mode or people will play just one mode to get higher on the leaderboards, defeating the purpose of omnithlon.
The problem with pp balancing through some kind of equivalency rate is that suddenly, if all modes are worth the same, then why not just play the mode you're already used to most? This also defeats the point of omnithlon.
The solution is then to weigh a player's total pp not just per play, but per game mode as well. For example, let's say a particular player has five 100pp plays in standard, five 100pp plays in taiko, five 100pp plays in ctb, and five 100pp plays in mania for his or her omnithlon top plays. The standard scores would then be weighted normally, the taiko scores would then be weighted normally and as if the standard scores didn't exist (so, from 100%), and the other mode scores would be done the same way. This means that if an omnithlon player plays a single mode for his or her omnithlon score, the player would only be able to get so far. However, if the player then starts playing another game mode to add onto their omnithlon pp, then the new scores from that game mode would be weighted from 100% and added on top of their current pp.
I think this would really encourage variety play, since that would be the only way you could maximize pp in this game mode. Playing a single mode in omnithlon would leave you at around 1/4th pp of omnithlon players who play all four game modes.
2. Players may want to make a particular play count for their osu!omnithlon score when they don't have the mode switched on.
The way this is handled is ultimately up to peppy, but I think it just comes down to two options.
The first option is to, instead of make omnithlon a mode you have to select, just make it an option somewhere that you want to make this score count for your omnithlon score. This would let people who play the game more casually not get frustrated by having to FC the map twice, and I like this option best as it avoids other possible frustrations the players might get if the second option is chosen.
The second option is to stick to the idea of how NF is handled, and give the user a penalty for submitting a score for omnithlon while having a different game mode selected, but give them the option anyway at the score screen. You could go to the extreme with this and just prevent the score for counting for omnithlon if that mode isn't selected, but then you would have to FC a map twice if you like to polish both your main ranking and your omnithlon ranking.
I think this is a good place to start when trying to figure out how to deal with players not wanting to have every score they play be submitted and also deal with pp imbalances between game modes. It does not address the problem of players wanting to play omnithlon but only play one or two modes as opposed to all four, but I think that defeats the purpose of omnithlon anyway. It should have the best all-around players at the top of the leaderboards after all, so they would have to be proficient in all four game modes.