> There's a 'most northern' point on the earth, and the surface is arguably continuous.
This was Hawking's point of view. One way of thinking about special relativity is that time is at right-angles to space, and everything travels at the same 'proper' speed (c) through this 4d spacetime, just in different directions (hence why "fast" objects experience "time dilation": their 4d heading is pointed largely in a space direction, so they don't progress much in the time direction). This extends to general relativity, which describes how these directions "bend" into each other. For example, the reason objects fall due to gravity can be described as spacetime curving such that "future" points slightly "downwards" (and hence slightly less in the usual time direction; which, again, causes time dilation).
AFAIK Hawking's idea was that the curvature at the big bang was so high, that "past" is pointing completely in a spatial direction; and hence there's no such thing as "earlier" anymore. Similar to how "north" near the north pole is at right-angles to "north" at the equator.
There's a 'most northern' point on the earth, and the surface is arguably continuous.
So it seems fine to have an 'earliest' point in time and still have continuous space-time.