How is a father able to find a game that suits these requirements, while maintaining the depths of the gamplay?
Not a father, though I've been going through 12 hour work-days quite often lately, so I guess time-wise it's comparable.
Unfortunately, from my experience at least, MMOs are not compatible with a lifestyle that implies a lack of free time. Most have several progression systems that span over large periods of time, and the game will always stick in your view people higher on the progress ladder than you.
You can of course have fun in them, just as Sarn says, however you do need to completely forget about comparing your character with that of your student - unemployed - bachelor friends.
On some games (including WoW, partially), it is possible to keep up with players that play much more, once you reach a certain stage of your character's advancement. For example, if you get to max level and gear yourself for the raid your guild is currently running, you can then remain on-par gear wise "just" by sticking to the raid schedule. This can sometimes be hard as well, however it usually implies 3 - 4 evening hours 3 times / week, so it is doable. If not, you can also look for guilds that match your schedule better.
Even so, you won't be able to keep up in other areas as well at the same time, such as gold-making, achievements, PvP, crafting, etc. So you have to find an area that interests you and focus on that one.
Otherwise, if you do not wish to focus only on one aspect or play the game in your own rhythm, you may need to look into other genres.
Single-player games are a good suggestion, as you can chew them in byte-size pieces and they also provide a constant fun experience, with little grinding downtime.
There are options on the multiplayer side however. Quite a few actually. "Lobby multiplayer" games have the ability to offer instant-fun at their core. You'll always be able to jump into a CS:GO / Call of Duty / Battlefield, DayZ / Arma 3, Hearthstone, LoL / Dota2 match and have fun. In many of them you would actually be on the same level with your hardcore friend as well, although the actual skill level may be different, which is something unavoidable as it comes through practice.
These also plateau much quicker, so even if there is meaningful character progression in them, it's usually much easier to reach the cap.
As an MMO fan, I''m still looking to find the right MMO and also play it the right way, although I have been drawn lately into the Lobby games a bit, especially through Hearthstone and a few shooters.
P.S. Guild Wars 2 is a great MMO for this type of thing, since it has almost 0 vertical progression once you reach the level cap. You continue to enjoy the frequent story updates, kick ass in WvW or just play instances PvP, where I believe everyone gets the same gear. It's almost like it was designed especially for fathers.