From that wikipedia article, delta is the ratio of y variance to x variance. If x variance is tiny compared to y variance (often the case in practice) then will we not get an ill-conditioned model due to the large delta?

If you take the limit of delta -> infinity then you will get beta_1 = s_xy / s_xx which is the OLS estimator.

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...