Then why all the brouhaha about not seeing yourself while messing with time, if this is doable?

Why, to ensure that nobody else does it! The brouhaha could easily have been stirred up by Voldemort himself through any number of means in order to further restrict the Good Guys' options.

Of course, it could have been stirred up by the first wizard to discover the power of working with yourself, and it's been passed down for so long that it's accepted without question, and Voldemort was either mad or foolhardy to have ignored the dire warnings. It just so happened that the dire warnings were wrong - Voldemort is incentivised not to correct them.