What do you call a Swiss man?

There are in fact very few cases where the customary demonym is root + -man— those you have named plus Irishman, Norseman, Welshman, Scotsman, and (obsolete, now considered offensive) Chinaman, and maybe a few others in Britain like Yorkshireman or Cornishman.

In the absence of a more established form, the demonym is usually the same as the adjectival form. Just as we would speak of an Indonesian, an Omani, or a New Zealander, we would speak of a Swiss.

It sounds abrupt, even to this native speaker, not only because it is monosyllabic (e.g. calling someone a Japanese also seems off), but because in today's politically correct age, referring to someone solely by their nationality or ethnicity is potentially problematic. It would be preferable to use the demonymic adjective in conjunction with a noun; Swiss man would suffice.

A man from Switzerland is called a Swiss man.

You can tag the nouns: male; man/men; gentleman; guy/s; woman/women; girl; boy; baby etc. to any nationality. And rightly so, otherwise phrases like:

"He married a Swiss"

and

"The other day while travelling into France by train, we talked to a very friendly Swiss"

are needlessly vague or ambiguous.

Moreover, Swiss man should not be spelt as one word. Some other examples are an Italian man, a Brazilian man a Chinese man. I tend to spell English man as two words but it can be spelt as one word, likewise Frenchman and Frenchwoman are commonly used names.

The words Englishman, Englishwoman, Frenchman, Frenchwoman, and Scotsman are nouns in their own right.

Even the Swiss news use the term Swiss man themselves.

Prince George's website swiped by Swiss man

I'm Swiss and this has never occurred to me until I read "a Swiss" in an article earlier today. It really does not sound right. If Shakespeare really referred to them as "Switzers", that seems good enough for the rest of us. My suggestion was going to be "an Helvetic" or "an Helvetian", derived from the latin name of the country (Helvetica, which is why the initials of the country are "CH" and why the currency is "CHF"). It may admittedly not be intuitive or obvious to readers, so perhaps this suggestion isn't the way to go.

Vladimir Nabokov and Shakespeare went with Switzer; one need look no further.

GERTRUDE     Alack, what noise is this?

CLAUDIUS Where are my Switzers? Let them guard the door.

Hamlet 4.5

(Interestingly, the Russian word for "doorman" is швейцар ["shveĭt͡sar"], perhaps from this usage. But I'm fairly sure "Switzer" indeed means "Swiss guy" in English.)

...with some weird light effects anon, when the torch-bearing Switzers are sent to find the body.

Bend Sinister 112.