For some things maybe, but usually there's an option where the risk is pretty negligible. I feel pretty confident we're not going to declare wooden spatulas to be a massive health concern.
There's a risk for wooden ones that are glued, specifically bamboo, or finished with something toxic. You should probably stick to ones made from a single piece of hardwood and are unfinished.
There's also a risk that any cracks will fill with bacteria.
TLDR; unfinished wood that is rinsed and dried on all sides will naturally trap and kill bacteria as it dries. Any finish interferes with this process.
Plus the risk of the alternative you pick being much worse than the original option.