My short answer: I don't think it's worth the cost and commitment.
Long answer:
I also grew up learning Suzuki violin. We were introduced with a small recorder (at 2, 3, 4 y/o). We were taught, by ear, hot cross buns, twinkle twinkle and a few other simple songs. We can do this ourselves, as parents, especially having music in our background (though it isn't a terrible idea for adults to learn on a recorder either ;).
With my kids (age 2 and 3), I expose them to music by playing what I am passionate for. We sing, drum and dance. I'll let them sit on my lap and help me play my full sized violin (they get to bow while I hold the violin). We practice on a penny whistle. When our buddies come over, young or old, music is often a part of "itinerary".
My three year old is getting a 1/16 violin this year for the holidays and my plan is to begin her lessons myself. We get our social activity through other means, and I simply can not justify the money when I am able to teach her (at a beginning level) myself.