Well, actually, if they side with with Prussia, they could get Savoy out of the deal. But, obviously, Switzerland wouldn't join Germany; and Prussia could never force Switzerland into the fold, not with Swiss defences as they were.
Savoy could have been at least administered by Switzerland OTL. Some elements in the 1815 Vienna Treaty
obliged Switzerland to administer Savoy in case of the royal house dying out.
The Swiss weaseled out of their treaty obligations as it actually happened, due to popular sentiment and to improve the relations with France.
The neutrality idea is actually far older than the direct democracy - it goes back to a battle in Upper Italy when both sides used Swiss mercenaries. Then the neutrality got into positive feedback loop as the Swiss benefited from the neutral position in the 30 Years war (selling food and weapons to both sides). The neutrality was finally officially recognized by the rest of Europe at the Congress of Vienna 1815, and the positive feedback loop started again.
The direct democracy movement OTOH didn't start until 1860s.
If the Swiss had any desire to keep territorial possessions outside of the core country as we know it, they wouldn't only get Savoy - also Veltlin, and Mülhausen were associated with Switzerland and could stay in. And then there is Vorarlberg...