I think it is a big deal. In many countries, you see a centre politician do some very normal thing that gets disingenuous outrage from the right solely with the intention (and effect) of dragging the Overton window over. From there, it is lather, rinse, repeat. The biggest reason for this is because capital has most of the power, most of the news outlets and capital is, of course, conservative.

When things do swing the other way, it is muted. Can you say Biden is more left than Obama? Were Harris' policies to the left of Biden's? Is Starmer's labour party more left than Corbyn's? On the whole, no.