As someone who first picked up a statistics book nearly 50 years ago, I might suggest that the prevalence of the first representation arises from the historical development of the field before high powered computing became available. But it also might represent a pedagogical advantage in terms of presenting the basis of statistical testing.
ANOVA means ANalysis Of VAriance. In the first representation, the components of variance in the observations are clearly the variance due to factor effects and the variance due to the error term. If you pick up an introductory practical statistics text from 50+ years ago, you may be astounded at how much space is devoted to the mechanics of using a hand calculator to calculate sums, sums of squares, etc., to get the values needed to perform an F-test comparing the magnitudes of the factor-related and error variances.
Since then the second representation, as a linear model, has shown its advantages both in computer execution of algorithms and in generalized models. For you this is a more heuristically valuable representation in terms of fitting and getting parameters.
The first representation, however, might make it easier for a new student to grasp the fundamental issue in this type of statistical test: are the effects of the factors of interest sufficiently larger than the residual errors? I can't say for sure, as I haven't tried to teach introductory statistics. Nevertheless I must confess that, despite a good deal of familiarity with linear algebra, I still look to the first type of representation when I get confused about about the comparison I am trying to understand.