A phrase meaning trail blazing without being concerned about the consequences to others

Solution 1:

A very obvious metaphor:

Their scheme was being railroaded through.

railroad [verb] [transitive]

...

1b: to push through hastily or without due consideration

[Merriam-Webster]

The active trampling of those in the way

railroad [verb]:

compel by coercion, threats, or crude means; synonyms: dragoon, sandbag

is mentioned by Dictionary.com.

and the expression 'railroad/ed through'

• Unfortunately, this scheme is being railroaded through Westminster under the pretext of immigration control.

by Cambridge Dictionary

To fit with OP's example, we can use the verbo-nominal multi word verb (and avoid a mixed metaphor):

They're railroading their way through, blasting aside anybody in their path to success.

Solution 2:

They bulldozed a path/their way through the objections

MW:

Definition of bulldoze one's way

to move forward while forcing other people to move out of the way

They rudely bulldozed their way through the crowd.

—often used figuratively

He bulldozed his way to the top without regard for people along the way.

bulldoze verb

transitive verb

1: to coerce or restrain by threats : BULLY

2: to move, clear, gouge out, or level off by pushing with or as if with a bulldozer.

3: to force insensitively or ruthlessly

intransitive verb

1: to operate a bulldozer

2: to force one's way like a bulldozer