Totally agree, this circuit is very sensitive to the layout construction.
Otherwise, I've seen many commercial VSWR meters using this kind of circuit, which should work if is build in concordance. Bridge type VSWR meters are better.
Higher impedances will provide higher voltages into the circuit, and these voltages are harder to deal with. I would recommend to stay on low impedance values.
LEDs do not detect high frequencies due to high junction capacitance (I didn't try the latest LED types), so you need the 1N60 diodes in place.
If want to use LEDs, best idea would be to replace the meters with LEDs, and adjust the series 1k resistor for proper work.