"...dark matter indeed interacts with gravity,..."
No.
Gravity is an interaction between mass fields or more precisely their space deficit aspects.
A thing (DM) can not interact with an interaction ('gravity').
DM could interact gravitationally.
The problem being there is absolutely no evidence of hypothesized DM responding to any visible matter's mass since nobody has ever seen/detected the stuff. or found DM additions to visible matter's mass.
If DM's mass field(s) don't respond to observable mass fields how could it be matter?
Dark Matter's mass field WOULD respond to visible source mass fields
IF, repeat IF DM were actual matter.
Put up or shut up, DM acolytes.
There is only visible matter & light responding to some mass with no obvious, identified source.
While associating mass with matter is reflexive keep in mind black holes have mass, but one would be hard pressed to say those have actual 'matter'.
Gravity waves are themselves distortions in space (like mass fields) that have been dislodged from their original source(s).