The circuit "issue_with_Vground.asc" has almost no connections to SPICE Ground.? The only connection to Ground is the input source, V2.
As you may or may not know already, SPICE (not just LTspice but all SPICE programs) requires a DC path from every circuit node to Ground.? From your "Vp" node, where is that path?? If it doesn't have one, either SPICE complains mightily, or LTspice tries to "fix" it for you by adding its own resistors to Ground, or it fails.? In your case, I guess it fails, with Singular Matrix errors and such.? ("Singular Matrix" means that the circuit matrix that holds the current or voltage values is singular and can not be inverted, but inverting that matrix is necessary throughout the simulation.)
If you change "tran 1m" to ".tran 1m UIC", the simulation can "run", and SOME of the simulated voltages are actually close to correct.? But nothing is really right.? Lots of complaints from the simulation in the error log.
Andy