I have tried oxyclean, vinegar, and alcohol. All leave spots.
It leaves spots mist likely caused by the water mineral deposits.. tap water.
Clean again but only use distilled water.
Baking soda. Vinegar...
This looks like an inherited piece that may of may not have been:
A. Originally off white but mistakenly spot cleaned with Undiluted Bleach
B. Cleaned with a wax furniture polish or
C. Treated with a spill proofing, and the Spill Proofing has Dated, Worn and grown splotchy.
From the picture I see a slight sheen here and there and find B or C likely but your guess is better than mine as you know the history and background of the piece.
So you have 2 options:
1. A. For Sealant Treated, (Spill Protected) Cloths, that are True White rather than off white or vanilla you need to use some Bleach as an Additive to Oxiclean, about a Capful in the Entire 7oz. Bottle, and Mark that Bottle 'For Furniture Use', because the Oxiclean alone does not ever do the Job on aged 'Spill Protection' nor does any product out there prevent aging.
Keep in mind Oxiclean Chemistry is very close to Hydrogen Peroxide, so: firstly, If peroxide will not do the Job, neither will Bleach with Oxiclean. Bleach with Peroxide msy work as an alternative, but gives off an Odor and Fumes that merit a Mask and Ventilation. I am sure Oxiclean and Bleach merit same safety precautions.
2. If or when it is not treated with a Sealant, you instead need to use a Garment or Carpet Suitable Detergent and whiteners rather than bleach (unless you want to take the color out) or concentrated cleaners [as Spray 9].
To accomodate this with a treated piece You can attempt to remove any Splotchy Sealant with a Furniture Safe Sealant Remover (alot like Paint Thinner or Paint Brush Cleanser).
I cannot refer any removers in specific but know that 'Spray 9' takes paint and veneer off walls and that may assist an endeavor of Removing or lightening or balancing Splotchy Sealant provided you apply with a Cloth rather than Spray it on. Again wear a Mask and Ventilate.