ToughSF: Anti-Orbit Laser Submarines
toughsf.blogspot.com
toughsf.blogspot.com
https://en.wikipedia.org/wiki/ASM-135_ASAT
You only need to pop up to 300 miles, hang in space for a moment, and let the satellite crash into you.
Trying to shoot down a satellite in high orbit (say a GPS or geosynchronous satellite) takes a lot more energy, in some cases more energy than it takes to get to LEO.
It also gets it wrong that orbital spacecraft would have free maneuvering. Making big changes in an orbit takes a similar amount of impulse as it takes to get into orbit to begin with. Ion drives, solar sails, and such things get much better specific impulse than conventional rockets but they all have low acceleration which will be unacceptable for wartime use close to Earth. Nuclear Thermal Rockets perform a little better but not that much better.
There is an old Heinlein quote to the effect that once you are in orbit everything is easy. It just isn't true:
https://caseyhandmer.wordpress.com/2019/10/02/there-are-no-g...
In the article's scenario, that's 300 miles and an indefinite timespan where you're being continuously burned by a megawatt to gigawatt-class laser that can penetrate meters-of-amor per second.
Most likely it hits you before you know what hit you.
For a speculative article like this, spacecraft drive serving a story narrative will need to move human-scale payloads (hundreds or thousands of tons) in human-scale timespans. (Weeks, not years) Therefore, gravities of acceleration, and buckets of delta V. Trajectories planned by pointing at a screen and saying "go there", not waiting six months for the right rocks to line up to give you a gravity assist.
Fusion or antimatter rockets, not contemporary technology.