C560 Captain reported an unstable approach condition due to a late ATC approach clearance. ATC directed a go-around and provided vectors for a subsequent approach which resulted in a safe landing.

Narrative

We were being vectored for the RNAV (GPS) Y Runway XX at ZZZ; the second in command was flying. ATC had us on a 160 degree heading at 2000 ft. MSL and less than a mile from the final fix when we were cleared to intercept the approach and that we were cleared for the approach. The FMS failed to sequence properly and the Autopilot did not capture the approach course or the glide path. I then took the aircraft from the second in command (SIC) and was maneuvering the aircraft back on the approach course and glide path. The aircraft descended below glide path; the Tower advised us that we were to the right of course and that we were low on the approach. We advised the Tower that we were correcting for the deviations when Tower advised that they had a low altitude alert. At that point we had just as we broken out of the overcast when the Tower told us to go around. We were then vectored for the ILS and landed uneventfully.The take away this: 1. We should not have accepted the approach from the vector that ATC had given us in that we were to close for the aircraft systems to properly sequence on a RNAV approach. 2. I should have discontinued the approach and gone around earlier; but I felt that we were sufficiently on course. Once I broke out I had the runway environment and was in a position to continue the approach but we were instructed to go around and I felt it best to comply.Bottom line: I allowed myself to be in a situation where the approach became de-stabilized very quickly and I should have immediately discontinued the approach and initiated a go around. Secondly; with the vector we were getting I should have advised ATC that I would not accept the approach and requested another more appropriate vector.This is not the first time at ZZZ that ZZZ [Approach] has vectored me in this this and they do not seem to understand that; especially on an RNAV approach; appropriate the sequencing of the equipment is imperative and these close intercepts can be problematic.