Pete,
A slippery slope fallacy occurs when one claims in the absence of any sort of evidence that a relatively small first step will inevitably lead to a chain of related steps culminating in an overall negative effect.
It is not a fallacy when one can provide evidence for or otherwise demonstrate that progression.
Are you aware of Biden's thirty-six year history as a senator? Are you aware of the bills he introduced and voted on during that time? If you are, then you understand that he is not exactly a fan of handguns either. (To put it mildly.) If you believe the draconian measures he proposed for nearly four decades as a senator were reasonable and measured, I would really like to know why.
Are you sure you understand exactly what it is that Biden proposes now? It is not simply a ban on the so-called "Assault weapons." It is a ban on every center-fire semi-automatic rifle with a detachable magazine, which includes a goodly number of makes and models. Again, if you believe this is reasonable and measured, I would like to understand why.
It also is not a logical fallacy to point out when an argument is inimical one's point. You said that, "People will hurt each other with whatever means they have, if motivated to" (Emphasis yours) and that this understanding "...is behind the logic of locking up the kitchen knives if an argument breaks out" and "It also is behind the logic of gun regulation."
Hyperbole is by definition not logical, but if your reference to kitchen knives was along the lines of "Hide the knives," and this was simply your own dry humor, then okay. (I've got a dry sense of humor too, so I can appreciate that.)
Even with that clarification though, you have still advanced an argument that would apply with equal force to all firearms because there is not a single one that we could not hurt each other with if motivated to.
Open ended arguments are incompatible with the notions of reason and measure. If pointing this out is a logical fallacy in your mind, then I would say, "Physician heal thyself."
On a more personal note, I would happily have this conversation with you on my patio while filling your wine glass. I appreciate your willingness to discuss this topic and wish you good health.