So yes, Switzerland does accept the relevant clause(s) of FAR 61.3.
Although you are correct, this sentence may be misleading.
Under ICAO obligations, the Swiss have to (and will) accept an N-reg that is flown by an FAA licensed pilot. Or to be more precise: A pilot holding a license that is issued or rendered valid by the authority responsible for the aircrew licensing for that aircraft. For an N-reg, that's the FAA. (EASA is trying to get a grip on N-regs, N-reg operators and N-reg pilots being based in the EU but that's another matter - and postponed anyway.)
The FAA has issued a rule, FAR 61.3, that they will accept any ICAO conforming PPL for flight on an N-reg, but only within the airspace of the country that issued that PPL. And subject to a few other limitations which are not relevant here.
So by virtue of getting into that N-reg aircraft, your Swiss-issued ICAO PPL has now become a de facto FAA PPL. Which allows you to start the engine and fly that aircraft - with the Swiss treating you as if you were the holder of an FAA PPL. Within the limitations of the original Swiss license and the limits placed on you by FAR 61.3 obviously.
So it's not the Swiss that accept the clauses of FAR 61.3. It's also not the Swiss that make an exception to their own rules or anything. Rather, it's the FAA which implicitly validates your ICAO PPL for flight on an N-reg. So you're flying the aircraft on a license that is accepted by the authority (FAA) that is responsible for licensing of aircrew on that aircraft.