To argue semantics we consider the meaning of 'independent' in contrast to that of 'separate'.
There may be a stretch of the notion of three 'separate' branches of government linked via the Constitution to an alterior meaning of 'independent' branches.
The distinction is important whereas 'separate' refers in kind to physical space and 'independent' to activities.
That the three branches are physically separate, there is no argument. However, their activities are never completely independent, ergo always dependent in some sense.
This is the crux of the issue. The judiciary is acting 'independent' when in fact they are merely separate, yet dependent on Congress for their role and jurisdiction.
Still we wonder what Tom Delay meant by his remarks and why he feels compelled in any sense to appease his accusers.
I think you missed the co-equal part of the argument.