Does Frozen Fortress put a trigger on the chain even if there's no units there?

Yes. Frozen Fortress puts its trigger on the chain at the start of each Beginning Phase regardless of whether any units are at that battlefield.

The trigger condition is "At the start of each player's Beginning Phase" — a pure timing condition with no additional check for units (Rule 383.2.a). Once that timing point is reached, the triggered ability becomes a Pending Item on the Chain (Rule 383.3).

When it resolves, the instruction "deal 1 to each unit here" simply finds nothing to act on and does nothing (Rule 359.3.e.6). But the trigger was on the chain and did resolve.