First, it will not be the servicers agree, but rather the owners of the mortgage notes. Second, the servicers for the most part are a bunch of guys who open envelopes, deposit checks and call up folks who they think can be induced to speed up their payments. They don't have any assets and will quite happily go bankrupt. Third, the servicers are not the cause of any damages. Once it is headed for bankruptcy, the contract has in effect been broken and all that is left is to pick up the pieces. You can be sure that any settlement agreements in all of this will discharge all parties of any liability for the losses incurred by reaching the new accommodation.
I don't think you are accurately describing the big mortgage servicers like Wells Fargo, BofA, Countrywide.
Once it is headed for bankruptcy, the contract has in effect been broken and all that is left is to pick up the pieces.
You are familiar with Section 1322, aren't you?
"modify the rights of holders of secured claims, other than a claim secured only by a security interest in real property that is the debtor’s principal residence"
You can cure the default through the plan, but you have to start making the regular payments if you want to keep the house.