The House and Senate might go along with some cuts to gov't positions, but broad, deep cuts will be opposed because there are too many jobs at stake. Today's Red district is tomorrow's Blue district if you start putting people out of work. Some agencies will backfill cut GS positions with contractors and now you're paying more over the long haul.
It depends on how and when it's done.
Ripping the band-aid off early is best (next Federal election is 2 years out), but may also require codifing some changes. Ex. No one is going to miss 87,000 IRS agents, but that reduction also needs to accompany a simplified tax code.
We did without the Dept of Education prior to 1972. It brought very little tangible value to education in this country as test scores continue to fall. Remove this layer at the Federal level and allow it to be handled at State and local levels. This may require additional processes at those levels, but it's not a Federal role.
The same can be done with much of DHS and so on. Just look at the bureaucracy operating our space program as opposed to what SpaceX has been able to do.
That's not to say everything should run like a private organization, but there's absolutely a ton of opportunity for efficiency to be gained, even if it doesn't result in the elimination of a full agency.
Will there be resistance? Of course, especially from those sucking on the teet of the Federal government and, probably, some activist groups, especially environmental groups who want studies on everything under the sun in an effort to slow or kill progress.
This will require a mental shift by many. There's certainly a balance to be had, but after some adjustment, my guess is we'll find eliminating many of these inefficiencies doesn't impact the daily lives of most people in a meaningful way.
My concern is ensuring those savings are not squandered. Some must be returned to the people and not simply reallocated to another government agency or program. If people see reduced tax burden, reduced fees, etc., then it becomes real.