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You're absolutely right, assuming the craft is on the L1 saddle point. The craft can, however, sit slightly sunward of the saddle point in a halo orbit. It wants to fall towards the sun (and enter a solar orbit) due to being on that side of L1, but you set it in the position it needs to be to balance the force of sunlight. There will be quasi-stable points in a halo orbits around the sun-facing side of L1 which could sustain a whole lot of these buggers.
KSP is great, but it only does two body physics (unless you're using the Principia mod -- never tried it). So you cannot simulate things like lagrange points there. The patched conics are a great first order teaching tool though, and KSP is great for that!
I'll take your word for it then.
Would have been a nice addition to KSP 2 if they hadn't fucked it up. I'll check out that mod at some point though.