My point is that the DeLucas stopped looking for the Mysterious Six because they couldn't find any leads on them iirc. They couldn't find a thing for a long time so they eventually stopped.
Now, the DeLucas know mc's father. Not in an abstract sense (for example "he's the guy that delivers pizzas on weekends"), they know who he is. They know his face and his name (or at the very least, an alias) and presumably more since he was involved with Rina. That's a lot to go on if you're hunting someone.
That's why I find it unlikely that mc's father had anything to do with Luna's kidnapping or the Mysterious Six. If the DeLucas knew (or even suspected) that he was involved, he would either have been found by the time the story takes place (and dealt with appropriately) or they wouldn't have stopped looking for him.
We don't know anything other than what has been said told to MC by the MC, but I don't believe just because the DeLucas know MC's father's name/alias, it's a 100% sure guarantee that MC's father would have been caught.
As said, Mysterious Six is an extremely elusive organization, and I think even with that knowledge by DeLucas, the organization would have the resources to hide him and perhaps even give him a new identity (if we assume MC's father is one of the Mysterious Six).
If MC's father went around carelessly in plain sight, and/or had no resources to back him up, then perhaps he might have been caught.
But with Mysterious Six backing him up and he himself being very careful, making himself seem like he disappeared? I'd imagine Myst6 would make it impossible for even DeLucas (or anyone for that matter) to track him down (until MC somehow gets involved, him perhaps being the unknown dark horse that even Mysterious Six didn't take into account).
As to "they wouldn't have stopped looking for him", well that really won't matter if they couldn't find him then, and the DeLucas still can't find him even to this day (and not like Wilfred would tell MC, "Your father is responsible for what happened to Luna and we're tracking him down." Wilfred, being extremely secretive, I'd imagine he'd keep that one pretty close to his chest).
MC's father, with Mysterious Six's resources in his control, perhaps chooses to remain a ghost until he finds it is the right time to strike at DeLucas and let his presence known once again.
To add, game does make a point in one or two instances of the MC whether by thought or phrase shows the whole "sins of the father is not the sins of the son." with MC saying things like "I am not my parents." etc so if the game goes MC's father is a member/leader of the Mysterious Six, and is the main antagonist of the game, well this theme could be explored much further I'd imagine.
With people "knowing" MC's father, who really knows him that well in the DeLucas anyway?
Only sure ones are Cordia, Wilfred, and of course "Serpent", and Gildart (since he's Wilfred's star pupil, and Wilfred let him in on watching MC, and likely the secrets behind MC's parents)
Probably also Antonio, since he's within the Higher ups inner circle (being Underboss, i.e. Capo Bastone and all), even though he's probably only been told who MC is by second hand account.
Rest of the soldiers, most of them (even very high ranked ones like Capos or Wilfred's Division) are currently pretty oblivious to who MC is and what makes him so special to Cordia, Wilfred and the higher ups (only possible exception is Angelo, who wonders why MC looks a little bit familiar, so he might catch on in the future).
In the end, everything I said above is all just fan speculation.
I could be right, or I could be wrong, but whatever actually happens, I'm not dismissing that possibility (of MC's father being evil, and is involved with Mysterious Six) until the game's story tells me otherwise.
