At the beach, I'm a member of the Bay Club.  I'm allowed to play the other two Troon managed courses -- Bear Trap Dunes and Bayside -- four times a season for a cart fee and unlimited times in the off season.  We also get discounts to play at River Run.

Walking is allowed anytime at all three clubs, including Bayside which has considerable hikes between many holes (I mention this because OP staff says that one of the reasons carts are required in the a.m. is because of the distance between 10 and 11 and 17 and 18.  They also claim that carts speed up play but studies show that's not true and actually slows play on cart path only days.)  Family membership is only $1300 if you pay up front and that includes the range.  

At home in Howard County, I play at the Columbia Association managed courses Hobbits Glen and Fairway Hills.  Both also allow walking anytime (and are VERY hilly) and Fairway has a half mile walk between the 15th and 16th holes.  CA would never discourage walking on these courses given their commitment to the health and well being of its members -- something that should be paramount for OP but obviously is not.  Many if not most of the members at the Bay Club are OP residents.  They left OP for difference reasons but not being penalized for walking in the a.m. is definitely one of them. 

I would like to come back to OPG@CC some day --  it is my favorite course at the beach and I want to support the amenities offered by OP and I help pay for.   In fact, my goal is to retire in a home on the golf course there within the next 10 years.  I hope by then walking will be allowed in the a.m. It also would be nice to have playing privileges at other courses with an OP membership.  Have you broached that with, for example, River Run?  I understand they're struggling and maybe would be amenable to such an arrangement.  

Charlie