The guillotine is a meme. We do not advocate it seriously.
The guillotine is associated with radical politics because it was used in the original French Revolution to behead monarch Louis XVI on January 21, 1793,
This is not true. It is associated with radical politics because of the revolutionary tribunal. And because of Robespierre regime, not in spite of it. Let me make a quick summary:
In September 1792 a popular riot took over prisons in Paris and killed everyone they found that looked like an aristocrat or a member of clergy.
It was considered to be the result of a popular sentiment, but was so chaotic that the authorities could not condone it, therefore it decided to install a revolutionary tribunal. Expeditious, but with some semblance of defense and justice. Danton was saying "We will be terrible, so that the people do not have to be" and "we will make a tribunal that is not good, it is at present impossible, but that is the least bad we can".
Danton and Robespierre, who had equal roles in the Terror regime, despite the popular narrative making Robespierre the bad guy and Danton the good guy, were moderates of that time. They installed it in order to decrease the amount of retributive violence.
When we, in France, praise the guillotine, it is more out of desperation and nostalgia for that time when powerful people had to defend in front of a biased justice but, for once, biased against them.
Later on, yes, it became the tool of execution of a variety of autocratic regimes. The Commune was an extremely progressist movement. When it burns down the guillotine, it is the death penalty in general that it rejects but it does not reject the idea of opposing with violence to the authoritarian government. But this time the government had fled so they formed a militia to attack them. Had them win, we would have seen another type of popular justice. Probably less bloody though.