A more measured approach might mean spreading the $1 trillion of cuts over at least a few years. To ensure that we aren't making things worse, we could strike a deal with ourselves; we cut however many tens/hundreds of billions could reasonably be accomplished in one year from the budget, but the cut would be conditional on not causing significant drops in important things like child health or people with homes. Perhaps temporary tax incentives could be involved, if that wouldn't be too icky. Depending on how well the goals are met, the next year's cuts are considered.
I understand that this all might seem rather centrally-planned to a libertarian. It also lacks the moral clarity of cutting $1 trillion immediately and eliminating five Cabinet departments in the first year. But it would keep from scaring away potential converts like me.