Why does a least squares fit appear to have a bias when applied to simple data?

If your model is different (y = Ax + b + e where the error e is not normal) then it could be that a different penalty function is more appropriate. In the real world, this is actually very often the case, because the error can be long-tailed. The power of 1 is sometimes used. Also common is the Huber loss function, which coincides with e^2 (residual squared) for small values of e but is linear for larger values. This has the effect of putting less weight on outliers: it is "robust".

In principle, if you knew the distribution of the noise/error, you could calculate the correct penalty function to give the maximum likelihood estimate. More on this (with explicit formulas) in Boyd and Vandenberghe's "Convex Optimization" (freely available on their website), pp. 352-353.

Edit: I remembered another reason. Least squares fits are also popular because they are what is required for ANOVA, a very old and still-popular methodology for breaking down variance into components (this is what people refer to when they say things like "75% of the variance is due to "). ANOVA is fundamentally based on the pythagorean theorem, which lives in Euclidean geometry and requires squares. So as I understand it ANOVA demands that you do a least-squares fit, even if it's not really appropriate for the situation.

If the dataset truly is linear then we'd like this linear estimator to be equivalent to the conditional expectation E[Y|X], so we therefore use the L2 norm and minimize E[(Y - L[Y|X])^2]. Note that we're forced to use the L2 norm since only then will the recovered L[Y|X] correspond to the conditional expectation/mean.

I believe this is similar to the argument other commenter mentioned of being BLUE. The random variable formulation makes it easy to see how the L2 norm falls out of trying to estimate E[Y|X] (which is certainly a "natural" target). I think the Gauss-Markov Theorem provides more rigorous justification under what conditions our estimator is unbiased, that E[Y|X=x] = E[Lhat | X=x] (where L[Y|X] != LHat[Y|X] because we don't have access to the true population when we calculate our variance/covariance/expectation) and that under those conditions, Lhat is the "best": that Var[LHat | X=x] <= Var[Lh' | X=x] for any other unbiased linear estimator Lh'.

In the wiki page, factor out delta^2 from the sqrt and take delta to infinity and you will get a finite value. Apologies for not detailing the proof here, it's not so easy to type math...