I agree with you: if you are going to have family time, the DO SOMETHING TOGETHER!!!!! Otherwise, in my opinion, you're building the idea that family time with the family.....bites. Blech! Who wants to sit in the living room around the TV all day????
We do have a family day: usually Sunday. Monday through Friday, there's homeschool, work, lessons/classes (soccer OR horseback riding lessons depending on the season, kung fu, kindermusic, and Bible study), and outings with mommy where I take the kids to a museum, the beach, a local bounce house place, a petting zoo, etc. On Saturdays, we play. But sometimes that's a soccer game, we do go to the fitness center together that day, it's often some birthday party to go to, or we'll have UFC fight night where friends will come over and we'll watch a fight and eat or whatever. It's often when we plan backyard bbq and the neighbors come over, etc. But that could change! We keep an eye on the community events and PLAN things that sound fun. (Hey, there's a seafood festival coming up....or a car show....or that new paintball place is opening up....the weather is going to be awesome, let's plan on going to the beach, or whatever seasonal events you have in your town, etc).
Sunday is our normal family day. We sleep just a little bit late (or I'll get up and get the boys their drinks and turn on a movie for them and my husband and I will lounge in bed a little bit), we fix a bigger breakfast (during the week it's cereal, yogurt, oatmeal, or frozen waffles but on Sunday it's a real breakfast: quiche, pancakes, eggs/bacon/biscuits, omelets, french toast and fruit, etc). We go to church at 10, are back by 11:15, have a very light little lunch and put the boys to bed for naps. When they're napping we'll either lay down for a nap or watch a movie and play scrabble or something similar together. When the boys wake from their naps, we may take a walk in the nature preserve, go to the beach, play outside something like bocce, croquet, soccer, Tball, tag football, or catch, or sometimes Jer will do a science experiment. Sometimes there's a chore that needs to get done that isn't a normal thing, so our older son has fun doing it with his dad (last week they went to "the man store"--Lowes, flushed the gutters on the side of the house to get the pineneedles out, and then made paper all by themselves). If that's what's happening, then I either cook dinner or our younger son putters around with me acting like he's helping too: we put the straw in a wagon, or he mows the grass with his bubble mower, so we're all together. Then I take our oldest to Awanas from 5:15-7:15 (a church club that is kind of a cubscouts meets vacation Bible school thing) while my husband plays with our youngest. Dinner is always ready before we go to Awanas and we eat it when we get home, around 7:30. Then it's the evening routine and getting ready for bed. We watch Amazing Race together too, that's our "thing" for years, before even being married.
Family day is important. But it's not just "a day" as much as "TIME". And we have never banned friends, neighbors, phones, or computers, and the TV is on at times but not a central part of the day or something we're expected to gather around. However, if you're busy "doing" or "playing" and having fun together, then the phones, computers, and TV do kinda get put off to the side. I wouldn't suggest going from "doing nothing" to banning all normal life and forcing "fun" on everyone. I say pick something to do and make your family time around that. It will just sort of grow and develop. Like, we just figured out that sleeping in to us is 7am (HA) so we have 3 hours before church to cuddle, cook, look at the paper and cut coupons, get ready for church, play with the kids. Church, then lunch, nap is at around 11:45 so they are up no later than 1:45 and we've got 3 1/2 hours of "whatever" before Awanas (which is negotiable---we like it, but if we had something great going on, we don't have to go). Routines and schedules are fun as long as they are still flexible and allow for life.