Hi,
highway=tertiary_link
is incorrect in this case. link types are for connections to the specified types of roads. slip roads/ramps and sometimes small connections on junctions fall into this category. And of course, they must be connected to a tertiary road.

Note that at least in context of Estonia, unclassified doesn't mean that it needs further classification. It's usually just a non-residential,  road which has a longer than 5-number ref number in Estonian road registry.