Star grounding typically makes worse oscillators than ground plane does. One can build an amplifier that oscillates, and vice versa, with either method.
There is no quick rule that can encompass all the considerations of a well designed circuit.
There are kernels of truth here and there: ground planes shield traces and star grounds maintain consistent DC potentials without creating loops. Most people -- the people who need "quick" rules to live their lives by -- take one or the other, and stretch it all the way to the absurd limit.
The key is to use each when appropriate, and to understand the equivalent circuit that one is creating in the layout. One must understand that the schematic is an idealized abstraction, not a description of the real layout. Conversely, one can model a real layout with an equivalent schematic, but it is likely to look very different from what one might be expecting.
Ground planes (at the PCB level) are generally not used for tube circuits, because the impedances are much too high. It is not possible to fabricate microstrip over about 150 ohms, while the circuit impedances are kohms at the lowest. It hardly matters anyway, as very few tube circuits need significant shielding, nor characteristic impedances. In any case, the low impedance of microstrip implies that stray capacitance is utterly dominant, and one should not exacerbate that by increasing it further (decreasing the PCB impedance).
Ground plane is still valuable, at the enclosure and cable level, to prevent crosstalk and interference. This, by the way, is where the myth of "terminate shield on one end only" came from: back in the toob days, electrostatic shielding is needed, again because the impedances are high enough that magnetic induction can be ignored.
Make no mistake: there is absolutely, positively nothing objectively good about leaving a shield open at one end, at radio frequency or with respect to controlled impedance conditions. You're literally introducing 100% noise to your signal, at RF!
The only reason it was done at all, was because the offending interference was electrostatic in nature, and the shield simply provides a path to shunt that field away. Likewise, not double-grounding the shield prevented ground loop currents from passing, typically a consequence of the only intense magnetic field source in a tube amplifier: the power transformer, and the currents flowing around the rectifier.
Instead, one should seek to extend the one truth (as it is at RF) down to DC, or as low as is possible. Avoid crossing current flow paths -- don't route the input around the rectifier, you'll pick up induced (electric and magnetic) hum or buzz. Do not ground coax signals to the chassis in one location, and ground the amplifier elsewhere; return the input stage to the same reference. Amplifier inputs and outputs are differential, just like everything else, by definition, because voltage is a difference. If there is a voltage drop along a ground return path, it is better for that voltage to appear on the output side than the input side, because the signal is larger on the output side. (So, if you are going to star-ground, return it to the input connection.)
It's easy to forget about AC and RF conditions. Star grounding is an invitation to oscillation. It's a series of resonant stubs just waiting to be excited; all it takes is an unlucky feedback path, probably some grids lacking adequate losses, or accidentally loaded with capacitance, and there you go. If each amplifier stage is referenced to its inputs, then one should obviously chain the grounds, from input, to stage 1, to stage 2, to output, to filter, to rectifier. That keeps all the loops precisely where they are needed (the loop between stage 1 output and stage 2 input is joined by a minimal length ground, and no currents cross it, such as rectifier-to-filter current, which is safely on the other end).
As ground plane increases capacitances, it can also be an invitation to oscillation, because of the increased capacitances. Maybe less likely? I don't know. The key, whatever the gross layout method used, is to prevent gain at radio frequencies, most importantly by adding grid stoppers. (Don't forget heater bypass if cascodes are used, too.)
Tim