The point of the branding yes, however the actual point of any right wing party in 2024 is not slowing change, but active regression.

Funnily, liberalism is the most politically aligned with not really doing anything, the whole philosophy more or less boils down to: we're not gonna get in the way of whatever you're up to as long as you're not getting in the way of what someone else is up to.

The left-wing is where progressive change actually comes from, and often you'll see resistance to actual progressive policy from both conservatives and liberals.

That should be the point, but in modern politics (especially in the US, but also in much of the western world from what I can tell), the people calling themselves conservatives have tipped over into reactionary/fascist territory, and the liberals have rushed into the void they left and taken over the "No change/slow change" position.

Originally they were trying to conserve the monarchy as democracy swept through Europe. Here's a video that goes into more detail.

In modern times conservatives courted racists in the US with something called the southern strategy. This has lead all of them down the path to fascism.