That does bring up an interesting question though: Had Doctor Who premiered back in 2005 and not had the previous series to back it up, would it have lasted?
I'm not sure it's possible to answer that, even hypothetically. The modern series is absolutely, in every respect, the culmination of the first 26 years. The 2005 series built on the concepts of the earlier run - aside from the Doctor and the TARDIS, and the basic format, you have the Daleks and (most importantly) regeneration showcased in that first revival series.
If the basic 1963 concept of the show had instead been created in 2005, it's probable that it would still be running today, but who knows. In any case, it wouldn't be the same.
Well the writing is strong and witty and fun enough that I think the show would still be popular, just not to the degree it is now.
It's that huge element of nostalgia and love for the character that is really sending if over the top. Even if you've never watched the classic series, you can feel how special and iconic the whole thing is.