Additionally, does that mean you always have to navigate to the daily note before inserting the time? (would be neat to do it with a small overlay from anywhere)
To automate this from anywhere in Obsidian, look at QuickAdd plugin, specifically its Capture functionality. You can use it to set up macro commands that you can fire up by pressing CTRL+P → your command, and it allows you to insert any text anywhere into any note, among other things. This Youtube tutorial by the plugin’s author goes into much more detail. You can then combine it with Dataview or anything else, really.
It’s a bit of a rabbit hole, but you can make very quick and powerful commands with this.