Oh, we could rebuild computers from scratch all right – technically speaking, that is. However, on the market side that would entail:
1. That we rewrite or otherwise replace every software still in use – not only off-the-shelf consumer software, but custom systems running inside companies;
2. That we come up with a definition of "security" all interest groups (industry and user groups) agree with.
Actually number (1) is somewhat feasible: start with a new market that still doesn't have a large legacy (say, mobile phones), secure a foothold, then slowly but surely eat back into the older, established markets. This is the strategy being employed by ARM on the hardware side, but unfortunately no successful mobile OS that I know has dared to truly break with the past and start with a clean code base on a modern language.
Number (2) is more problematic. We've seen some initiatives to get security bolted into computer systems from the ground up in the last years, but seemingly companies cannot resist to bundle some content management restrictions to the runtime security (hello DRM), so user groups tend to distrust them.
In the end I think the problem is economic: things mostly work, and insurances cover the costs when they don't, so there's little drive to improve. Perhaps if we embarked on a new Space Race, and the need to write complex and really trustworthy software increased, we could get this started?