On our website, parents are able to book a nanny. The parents are able to find the perfect nanny by using the search filter. One of the options we provide is the option to filter on a date. This is not user-friendly if you can't filter on a time. The nanny is able to fill in there time they are available in there CPT but the parent is not able to select a time in the toolset calendar. Is there an option in Toolset to let the parent choose a time like here: hidden link. Long-story-short: The parents must be able to select a time on a specific date by using the calendar or the other option you can provide.
I've added two images to support the current situation/problem.
I checked and you are right that even though you can create a date-time field with Types, when you insert a filter control for that field in a View it can only be used to provide a date, not a date and time.
It's a missing feature rather than a bug, but I'm going to escalate this to see if the developers will agree to add support.
If they do it will likely take some time in the context of the existing priorities, so let me give you a few pointers in case you want to implement such a feature yourself.
Date (and date-time) fields are stored as UNIX timestamps (e.g. the number of seconds since 1 Jan 1970). See hidden link
When you add a filter control for a date field to a View, there is an input which is shown on the front-end in text form (e.g. "November 1, 2019").
But if you use your browser dev tools you will see that there is a hidden input created whose value is the actual timestamp that will be used by the filter. See the screenshot from my sample site, with a custom date-time field "validity".
The value of that hidden input (1572566400) corresponds to 2019-11-01 00:00.
So to include the time you could use a custom JS solution whereby you add your own form inputs for the time, and use JS to modify the timestamp in the hidden input to add the number of seconds which correspond to the time entered in the input.
If there is something specific you get stuck on I can help, but I can't develop the feature for you.