The pipe going into the sump for the main sump pump is quite a bit higher than the bottom of the basin for the laundry sink pump so I don`t think the drain system is lower. It got me looking around and I discovered the float switch on the main sump pump is defective. I assumed it was working ok and it`s not. Thank you for that.
So, what you are saying is the drain line bringing water from the footers is higher than the bottom of the laundry sump, correct?
That leads to the question of what are the relative elevations of the bottom of the two sumps? Usually, the ground water sump also has holes in the sides and bottom of the sump to let ground water in to keep the water pressure from causing leaks into the basement through the walls and floor. So, like Twowaxhack said, one would assume that the "existing ground water drain system would be lower than your laundry sump pit." But if the bottom of the ground water sump is higher than the bottom of the laundry sump, then you may want to consider another option for the ground water.
You've apparently not had a problem until recently, but that doesn't mean that "repairing" the laundry sump to almost its prior setup will fix the problem. Again, as Twowaxhack said, "Even if the basin was repaired it still might push up from hydraulic pressure some place else or around the basin."
I know your ground water sump is located on your west wall and the laundry sink sump is on the south wall, but I don't know how far apart they are or if this is a finished basement. But as it is obvious that ground water is entering your laundry sump, you might consider piping the laundry sump pump to the same discharge line as the ground water sump pump. Then install a ventless laundry pump that connects directly to the sink outlet, as in the other post I mentioned, and route that discharge to the septic system. Actually, even with the cost of the ventless laundry pump around $250, this approach is very straight forward and probably would cost much less that any "repairs" would be for the existing setup. You would increase your capacity to pump ground water from under/around your house and assure that only sink discharge goes to your septic system. Seems like a win-win to me. For your laundry sink pump to run every 30 minutes, it means there is significant ground water entering your laundry sump. If the bottom of the laundry sump is significantly lower than the bottom of the existing ground water sump, you could adjust the pump float switches to come on later in the laundry sump to closer match the level the ground water sump is trying to maintain. No need to pump any more ground water out than is necessary.
For your consideration.