Unfortunately, the argument I had had in mind seems broken to me now, and I retract the claim. I wonder what of it can be salvaged?

Let me explain what I had had in mind. The main idea was this: it seems we don't need to flip the switches infinitely often, but rather only finitely many times for any given formula, based on the modal depth of the formula. 

And so I wanted to mimic that by weakening the notion of independence to $k$-independence. Specifically, define recursively a notion of $k$-independence
for a family of buttons $b_j$ and statements $s_i$ to be used as
weak switches, as follows:  for $k=0$, every such family is
$0$-independent at every world; the family is $(k+1)$-independent
at a world, if you can operate the buttons and switches to achieve
a new configuration (pushing any desired additional buttons and
setting the switches as desired) by moving to a world where the
family remains at least $k$-independent. Basically, a family is
$k$-independent at a world, when you can operate the controls as
you like $k$ times.

The observation was that if one has a lot of independent buttons, then you can easily make $k$-independent families of buttons and switches. Simply form disjoint groups of $k$ buttons, and for each group, let the corresponding assertion $s$ be the parity count of the number of them that are pushed. 

With such a family, you can flip the parity count by pushing one button, and so this family will be $k$-independent at the original world. 

My idea then was to adapt the usual button+switch proof, by folding in a $k$-independent requirement: the simulation lemma would work for formulas of depth $k$, provided the family remained $k$-independent. 

But that part seems broken to me now. I had thought at first one could rule out your counterexample by insisting that every node has independent unpushed buttons. This does in fact rule our your example, but it doesn't seem actually to enable the simulation lemma to go through. If the weak switches are not actually switches, then they could be possibly necessary in the potentialist system, and this will not necessarily be simulated in the Kripke model.