The existing values of crossing= have attempted to encode if the crossing is signalized or not, if it is marked or not, and various region specific implications of each. I’d recommend against further overloading this key with another attribute (whether the crossing path is continuous or not). On the other hand perhaps the addition of crossing=continuous or crossing=continuous_path would further increase adoption of crossing:markings= and crossing:signals= to explicitly tag these attributes on continuous path crossings.
I’d suggest a new key using the crossing: namespace. Perhaps focusing on tagging which of the two intersecting ways is continuous. Something like crossing:continuity={highway class} or crossing:continuous_way={highway class}. So at a crossing between a highway=footway and a highway=tertiary you could tag crossing:continuous_way=footway to indicate that the footway is continuous across the tertiary road. For a cycleway & residential street crossing where the cycleway is continuous: crossing:continuous_way=cycleway. A schema like this would be flexible enough to handle a continuous cycleway crossing a footway without ambiguity about which one is the sidepath/sidewalk. It could also specify that a highway=pedestrian is continuous across a highway=tertiary (crossing:continuous_way=pedestrian) or vice versa (crossing:continuous_way=tertiary).
If this level of flexibility doesn’t seem necessary, there could just be two values: one for “paths” (footway, cycleway, path) and another for streets/roads (residential, unclassified, tertiary, pedestrian, etc). crossing:continuous_way=path would state that a highway=footway is continuous across a highway=unclassified. crossing:continuous_way=street would explicitly state that a highway=tertiary is continuous across a highway=cycleway but this would be the assumed default and not necessary to tag.
** After writing this I’m realizing this idea could apply to any junction between two highway= ways whether is considered a crossing or not. So it could make sense for the key to use a highway: namespace instead.