Wilcoxon signed-rank test - the "distribution of differences is symmetric" assumption How do you define "symmetric" here? In a paired test scenario, if one treatment consistently produces better outcome than the other treatment, I'd imagine the differences to be clustered on the "positive" sign, without many "negatives". Wouldn't that mean the distribution of the differences is NOT symmetric (unequal amount of positives and negatives) if the H0 is false?

"Symmetric" means the population distribution of differences to the left of the population mean and median (assumed to be zero) exactly mirrors the population distribution of differences to the right of the population mean and median.
That said, (1) if the population distribution is symmetric, the null can be rejected if the differences are not centered on zero, (2) if the population distribution is centered on zero, the null can be rejected if the distribution is asymmetric, and (3) if the population distribution of differences is neither centered on zero, nor symmetric, the null can be rejected. How much not centered on zero, or how much asymmetric in order to reject the null relates to power and sample size.
Your case where "one treatment consistently produces better outcome than the other treatment, I'd imagine the differences to be clustered on the 'positive' sign, without many 'negatives'" is one where your sample distribution of differences is not centered on a zero mean and median, and would be evidence favoring rejection of the null. However, your example does not indicate whether the population distribution of differences (or sample distribution of differences, for that matter) is symmetric about its (non-zero) mean and median.
