If there are only a few items, yes - checkboxes are great for short lists. But if it is a long list, no - a huge pile of checkboxes is awful. So if your list contains more than a dozen items, just stick with the multi-select.
And you are right to just use the built-in UX the browser offers. Flawed it may be, but it is standard. Don't confuse your desktop users by inventing something new. They know their own browser.