Digging back a few decades, back to my years in typography, I don't think this issue is really "old school" versus "new school". Rather, I believe it is a matter of whether the text is typeset or simply typed.
With typeset text, spacing between characters varies based on the characters, based on an understanding of what makes text optimally readable. That's why we get ligatures (special presentations of two characters, such as 'f' and 'i', that combine the two into a special character, so that the top of the 'f' doesn't look strange right next to the dot of the 'i'.) I believe something similar applies with spacing (with, if my memory serves me, the typical spacing between a period and the beginning of the next sentence something more than one space but less than two spaces).
A typewriter, of course, didn't have any means of varying spacing between characters. All it can do is put in one space or two, so since one space is "too close together" for optimal readability, the standard became to put two spaces in.
Note that with HTML it doesn't matter. HTML knows that people are inconsistent in this regard, and so it ignores all instances of multiple spaces. This is "end. Begin" with one space:
"end. Begin"
... and this has ten spaces after the period:
"end. Begin"
No difference. Browsers do the typesetting for you, overcoming the variance between typists.