Are those restaurants making aioli from the scratch? Or are they mixing a garlic paste with bottled mayo?

I'm asking that because:

• you said it's "very, very white". Even eggless aioli is usually a bit cream-coloured.
• you said the taste is consistently the same across restaurants, but taste and texture change a lot depending on the person who do it from the scratch.
• it makes sense for restaurants as it's faster to use mayo as the base than to do it the traditional way.

With that in mind: oily taste in any emulsion hints the emulsion didn't form properly. Did you make sure you smashed the garlic into a smooth paste? Are you adding oil gradually, and making sure you emulsified all the oil you added before you add more?