Would it be faster to just get a mining laser and shoot asteroids than landing on planets and scouring for resource nodes when it comes to gathering materials like phosphorus and manganese?
Driving around on the planets for these things is maddening. They're so far apart, barely drop what I'm looking for, and there's never really any danger to break up the monotony.
Can I obtain these materials for engineering/synthesis from asteroids, or do those only drop the cargo space minerals used for trade?
Koli Discii C 6 is your best bet to farm engineering resources imo. Right in the bubble, easy to get to, and a variety of mats to pickup. You'll need to relog though to keep up the farm but you can get a decent amount of crap in an hour or two. You wanna find the crashed python on the planet.
Can't speak for synthesis mats though, never bothered with it myself.