If your dates are stored as date data type you must use data data type in your query.
Rather than
try with
'$gte': new Date('2018-01-01')
The field name in your query must also match a field name from your documents. If your documents have the fields time_start, time_end and last_updated, a query with the field name datetime will not produce any result.
It might be helpful if you could share the whole function where you insert and the whole function where you query.
I suspect the error is in the context of query rather than the query itself.
Also specify the programming language. I think it is python by the use of datetime.now() but a confirmation will be nice. You could tag the post with the language.
If you could run an aggregation that uses $type on your date fields to confirm the data type.