I am working on a the design system of a large system for users that use it on a daily basis, and I want to define the multi select dropdown. on one hand you can know in advance that you have multi select if you have the checkbox, on the other hand the design is cleaner without them and the user will figure it out after a few uses since he will use it every day
Checkbox is an indicator for users to quickly understand the multiple select option.
If your question is whether or not to use the checkbox, I think it looks better and does have a better user experience with the checkbox especially the unconscious overview it gives the user while making multiple selections.
I am still new to the field I don't know whether I am right or wrong but what if we tell the user how many selections he/she has selected then this way he might naturally guess that he can also select multiple items at a time. Like we can indicate that 1 item selected or n number of items selected.