You're literally saying "since a subset of the GOP fights hard, the Democrats need to compromise so they don't look bad".
They need to compromise with the non-tea party republicans. Republicans are more and more often (though still rarely) going to the left, to appease democrats, than to the right, to appease the tea party. The recent money sent to DHS is an example of this.
Progressives wanted to stop funding ICE entirely, other Democrats wanted to give funding to take care of only the kids, Republicans wanted to give funding to take care of everyone without being forced to release anyone earlier than they want to, and the tea party wanted to give funding to take care of everyone as well as give Trump wall funding. I agreed with the progressives, but do you see how giving funding to try and provide adequate care to everyone might be the best possible option?
They could take a stand and not give funding, but thousands of kids would pay the price. Trump wasn't going to let those kids, or adults, go if he didn't get the funding. Democrats in Congress can't fix the fact that Republicans, and their voters, are still defending their concentration camps. Letting Trump torture more and more people wouldn't have helped anything, but enabling him, even partially, won't stop it either.
There are already a lot of stories regarding costs. There were a few articles pointing out that the kids could get suites in Trump hotels for cheaper than the cost to detain them in the camps. There was a huge argument over funding because they ran out of money, and Republican voters, overall, blamed Democrats for not giving them enough money.
You're forgetting what the GOP is promising: The GOP is offering them free money (through tax cuts) and improved security/functionality of immigration enforcement. Your offer is just for better security/functionality/efficiency.
I think you're looking through rose colored glasses at the situation. I agree with what you said, but that is probably why nothing like it will ever appear in the largest media outlets. Clinton offered a plan to train coal miners in easier, higher paying jobs, at no cost to them, to replace their disappearing coal jobs. During her explanation, she said, 'coal jobs aren't coming back'. That is the only part that got any news coverage. I still think 90% of coal miners don't even know she had a plan.
That's what I meant when I said I have no idea how to solve the problem. I'm not saying we should give up, but shutting it all down because things suck, the thing that the GOP wants more than Democrats, won't bring comprehensive, positive change.