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?
It's been a while since I've played but I have this spreadsheet bookmarked as my go-to for engi mats. In my experience I had a lot more fun by farming tons of all the mats and then ignoring them. Trying to accrue them naturally takes forever.
The raw material notes in that sheet apply only to legacy. For live, it's the same planets, but you either drop in the blue and look yourself, or you navigate to some known coordinates. https://forums.frontier.co.uk/threads/live-4-0-materials-farming.610751/#post-10004234
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.
In case of Manganese (you'll have to do research for Phosphorus yourself), since it's common in raw group 3, I would go rather after Ruthenium and trade down.
You are looking for
EDIT: mining will also give you raw mats but rather lower grades and I don't know how to search for a ring so you get mats from group 3 and not another one