So-called "slippery slope" fallacy isn't that "any insinuation of a slippery slope or inevitable progression is fallacious". It's moreso multiple different fallacies that can occur when making a slippery slope argument, which, if done correctly, IS logically valid.

Here are some examples of fallacious "slippery slope" arguments, most obvious to least:

1.) "If we allow Bob to eat all of the peanut butter, we will no longer have peanut butter. If we no longer have peanut butter, we will go mad and kill each other."

This is fallacious because it is a non-sequitur; there is no explicit logic that connects mad murder sprees with loss of peanut butter, and there is no absolutism that that will be the outcome and not some alternative outcome.

2.) "If we run out of food in our home, we will surely perish."

This is also fallacious. The connective logic here is actually visible this time, though; it is true that most human beings will die of starvation if deprived of food for too long. What isn't true is that death is the absolute result of running out of food in our home; the alternative may be something like "we will visit the store as a result and buy more food".

Here is that last argument presented logically and soundly:

"If we run out of food in our home, we may perish."

It doesn't presume the outcome IS death, it just asserts that that is one possible outcome of that situation or event, which is correct. It's not likely you'd die of starvation if you have food, but it becomes more likely if you do not currently have food.

The argument made by this meme IS logically sound. It is not explicitly asserting that that IS the outcome or the only outcome, rather it's pointing to the fact that, by allowing the prior things, you are setting a precedent which would enable the final thing. You are enabling things closer and closer to that final one, which makes it more of a possibility, especially if not sufficiently resisted. It also harkens to the motives of actions; the idea being that, if people in power have motivations to impose the prior things, they (more) likely have motivations to impose the final.