How to balance academic reputation of PhD supervisor and teaching experience
I'd advise caution: completing a PhD with no teaching experience is risky, as you're putting all your eggs in the "research" basket.

As other answers mentioned, there certainly are jobs that don't necessarily require teaching experience.  These are likely to be jobs in places where research is paramount; probably the top-tier research universities.  There is massive competition for these jobs; the number of such jobs in such a year is at best a few percent of the number of new PhDs.  Only the most successful researchers will have a chance at getting them, and if you don't turn out to be one of them (or discover you just don't enjoy research that much), you'll be out of luck in this arena.

At most of the remaining institutions, teaching is more important and experience will be valued, and you'll be in competition with candidates who have substantial experience.  So if you don't have any, you may be out of luck here too.

(I don't share MHH's confidence that alternative teaching activities such as high school outreach will be viewed as similarly valuable by a search committee.  Community college teaching might, but it also involves a higher level of responsibility and time commitment.)

That would leave only the bottom of the barrel of academic jobs: those who don't care too much about research accomplishments or teaching experience, and just need a warm body with a PhD.  These won't be nice jobs: they'll have high teaching loads, low pay, and limited instructional support; and even so there's intense competition for them.  

The remaining option would be to leave academia for industry.

So in summary, if you aren't able to gain meaningful teaching experience, you'll be committing yourself to aiming for the most difficult jobs to get, without much of a fallback plan (within academia) if that doesn't work out.

(Disclaimer: My experience is limited to the US, and mostly in mathematics.  Things may be different elsewhere.)