"This guy" isn't exiting from a skylight, that's a roof access hatch specifically designed to give roof access. Also, there is often (but not always) a ladder to climb to get access to that kind of roof hatch.
Unless your skylights are hinged (a pic from the inside of your skylight, zoomed in close enough for us to see if there are any latches might help), then no, the only way to access the roof through the skylight would be to break the plastic/glass. Doing so would guarantee that's the source of the majority of your leak.
If you're that fearful of ladders/heights that you're not willing to climb onto the roof (is your roof flat, by the way, like the stock image you shared, or is it pitched), then I'd strongly suggest that you get someone else to climb up there to do the work. Being on a roof, for someone who is afraid of heights, is just a bad idea. You're likely to be so focused on not falling that you may fall anyway, and even if you don't, you may not do a good job of patching since you're not thinking about that, but about falling.
It doesn't have to be a contractor, maybe a family member or friend who is willing to help you out.