I know one thing at my parish that has brought many men into living their faith more fully is when they participate in some sort of renewal program such as Christ Renews His Parish (which perhaps reminds them of team sports with its team building approach starting on the initial retreat). The men who have been through that program (a new group of 36 participate every 6 months in English and the same in Spanish) tend to get more active in a variety of ways in their families and in the parish. The additional benefit to the parish is that they talk the program up to their buddies and coax or bully them into at least trying it and whoohooo, even more men getting active and taking ownership in their parish!
We have a large and very active “Men’s Club”, 7 male softball teams, an active group of Knights plus a “Brothers in Christ” organization that was started by a man who went through CRHP and saw a gap in programs for helping men to live out their roles as spiritual head of household, etc. (I believe our pastor recently said there are now about 100 different ministries and organizations but I listed some of the fun male-only ones.)
Our parish also has 24/7 adoration with only men or married couples together “allowed” to adore in the overnight hours for safety purposes. It really brought men out of the woodwork and dramatically increased the total number participating in adoration when we went to 24/7 from a more limited “daytime” schedule.
Even men who maybe were half-hearted about going to mass when they first arrived in our parish many times get a kick in the pants from the other guys while having a beer after softball or while helping to set up the carnival tents, etc. I think there are many ways to get men more involved in the parish and coming to mass that have nothing to do with excluding women from active roles assisting during the liturgy or in youth ministry, RE, etc. We have women assisting as readers, altar servers and EMHCs, but the numbers appear about 1/2 and 1/2 or even a little more toward the males just from “eyeballing” it.