Indeed, and even more, NASA uses the geocentric earth model to calculate the paths of all it sends into space, to anywhere.
No, it depends on the mission which central body is used in mission planning. This is because when in orbit around a central body, the central body's gravity exerts the greatest influence on the orbital characteristics mission planners are interested in. Calculating orbits can be somewhat simplified to a modified two-body problem with orbital perturbations. So for example, flight engineers with Parker Solar Probe will use the sun as it's central body (and one of the various sun centered reference frames - HCI, HEEQ, HGI, etc, etc...)
Multi-Mission Orbit Plotter | Parker Solar Probe Science Gateway
Parker Solar Probe
MAVEN's flight engineers will use an areocentric frame since it's in orbit around Mars (and so it may be MARSIAU intertial reference frame).
Flight engineers for spacecraft in earth's orbit, such as the ISS, will use an earth-centered reference frame (probably ECI or ECEF) since earth is the central body and the orbital characteristics around earth that are of interest.
When doing trans-orbits (going from say Earth to Mars) obit trajectories must take into account all the relevant bodies when calculating delta-V's for transfer orbits. So when calculating a flight from Kennedy Space Center to orbiting around Mars, one will have to account for the injection into earth's orbit, orbit around the earth, transfer to Mars, injection into Martian orbit.
When considering spacecraft attitude or instrument specifications, a flight engineer may need to use a spacecraft-body-centered or instrument centered reference frame and coordinate system.
Basics of Space Flight - Solar System Exploration: NASA Science
The decision of a central body and reference frames to use depends on mission specifications. In
all cases though, a heliocentric solar system is true because the laws of Kepler and Newton are true and it's Keplarian and Newtonian rules that must be used when calculating orbits, budgeting for fuel, calculating thruster burns, etc... All celestial objects, whether it be Parker Solar Probe, the ISS, the Earth, the Moon, or the star Sirius, follow the rules described by celestial mechanics and astrodynamics. And it's these laws that describe the earth's orbit around the sun.
So no, it's not true that NASA uses a geocentric model for spaceflight. NASA uses Keplarian and Newtonian dynamics. Missions will select reference frames based on mission needs, but a choice of reference frame does not change the underlying physics (in fact, the opposite is the case). To argue for a geocentric solar system where the earth is the central body with the sun and planets in it's orbit, you need to change the underlying physics, not merely change reference frames.
https://naif.jpl.nasa.gov/pub/naif/...ual_docs/17_frames_and_coordinate_systems.pdf
https://www.amazon.com/Fundamentals-Astrodynamics-Dover-Aeronautical-Engineering/dp/0486600610/ref=sr_1_1?crid=PA8XBOLBWG8S&keywords=fundamentals+of+astrodynamics&qid=1571249994&sprefix=fundamentals+of+astro,aps,196&sr=8-1