I'm not sure that any spaceflight missions to date have really needed SR/GR.
The Apollo missions finished in ~1972. GPS didn't kick in until ~1978.
The Hubble Space Telescope's Pointing Control System (PCS) uses guide stars for alignment. Interplanetary probes presumably don't rely on Earth GPS(!). The slingshot effect tends to be presented as a Newtonian exercise in momentum exchange, partly for simplicity, and partly because GR has some issues with velocity-dependent gravitomagnetic dragging effects, http://maths.dur.ac.uk/~dma0rcj/Psling/sling.pdf :
"... by its usage the slingshot effect is a modern triumph of
The fastest ever space probe seems to be the Parker Solar Probe, listed as having reached 692,000 km/h ( ~192 km/s? ) . By comparison, the the speed of light is more like 300,000 km/s So we're talking about less than a thousandth of the speed of light. At v<0.001c, differences between SR and Newtonian calculations are likely to be small compared to other errors (chemical rockets!). NASA don't expect their initial probe trajectories to be perfect, and the craft have manoeuvring thrusters to make changes and trajectory corrections during their multi-year missions.
GPS and GR
GPS calculations are designed to incorporate SR/GR corrections, but analogous effects also show up under Newtonian-based theory. The two "important" GPS corrections that GR folk like to talk about are the gravitational time dilation effect down on the Earth's surface (which makes GPS' orbiting atomic clocks tick faster than us) and the SR transverse redshift (which slows them down again). The clocks are typically adjusted to compensate for these effects before launch.
The "Newtonian" gravitational shift effect that changes the energy of light as it crosses a gravitational gradient was described by John Michell back in 1783 (paper). Einstein then seems to have rediscovered the effect and in 1911 pointed out that the unavoidable consequence was gravitational time dilation ("On the Influence of Gravitation on the Propagation of Light"), but for simplicity Einstein's 1911 paper derives the effect within Newtonian gravity. So while gravitational time dilation isn't usually "historically" considered to be Newtonian theory, the Newtonian version predates GR by a few years, and could and should have been a Nineteenth-Century prediction, and presumably would have been if not for human fallibility.
Earth gravity is almost certainly going to be probably too weak to tell the two predictions apart, because the Earth's gravitational terminal velocity is only about 11 km/s (as opposed to light's three hundred thousand km/s).
Newtonian theory also gives a counterpart to the second "relativistic" GPS correction, the SR transverse redshift effect. The Newtonian version of this is an "aberration redshift" caused by the forward deflection of rays, which causes the light entering a "transverse-aimed" detector to receive a ray that has a slight recession redshift component.
Oliver Lodge "The Ether of Space" p134-136:
"... a spurious or apparent Doppler effect due to common aberration. "
The Newtonian "transverse" effect appears identical to the SR effect other than it's a Lorentz-squared redshift rather than a single Lorentz redshift.
As a result there's no obvious reason to believe that GPS wouldn't work just as well outside the SR/GR context, and Ronald Hatch, one of the independent GPS technical experts who served on the GPS board until he died this year (2019), spent some years arguing this point.
In general, in practical situations, we use the Newtonian relationships for astronomy rather than the SR/GR set, because the Newtonian set are simpler, and the differences are so small at normal relative velocities that there's no appreciable difference between them:
"... because the velocities of planets, binary stars and clusters of
stars are hardly ever more than a few hundred kilometres per second,
the classical red-shift equation is accurate enough for most
Foundations of Astronomy, Third edition, Michael A. Seeds (1992)
If we ever manage to get spacecraft travelling at "full percents" of the speed of light, then maybe the difference might start to be important.