Is it possible to have such a visible column in SQL Server filled with automatic values, same as the hidden uniquifier column?
No there is no built-in feature to do this.
Or do I have to manually calculate such a column when I insert rows?
You could consider using INSTEAD OF or AFTER triggers to maintain the correct uniqueifier-like behaviour for the same-time group when rows are inserted/updated/deleted. This may require some careful thought to operate correctly under high concurrency, but it is probably best done as close to the data as possible (i.e. server-side, in the database, rather than client-side).
Alternatively, make the index unique and implement retry logic. Since the expectations seems to be that collisions will be rare, this might well be acceptable.