In one sense, yes, percentage losses have a greater impact than percentage gains. But while losses reduce the base from which subsequent gains can occur, they also reduce the base from which subsequent losses can occur.
Stock prices are often modeled using geometric Brownian motion, reflecting that a stock (or stock index) cannot go below zero. The process involves "drift" (average return) and "diffusion" (random ups and downs). If a stock had an equal probability of rising 50% or falling 50% in a given period, it would mean the drift (average return) is negative. This is not observed in historical stock returns. So indeed, "bad news makes the stock drop less (all other things being equal) than 50% here".
However, once you apply leverage, these safeguards go out the window. A leveraged portfolio can go below zero, and leverage can turn a portfolio with positive drift into one with negative drift. For example, if the stock had equal probabilities of falling 10% or rising 12%, that is positive (0.90 x 1.12 > 1), but if you lever it 2x so it falls 20% or rises 24%, that is negative (0.80 x 1.24 < 1).
EDIT: What I am calling "drift" here is the geometric average return, corresponding to
mu - sigma^2/2 in the linked article.