So when fitting a trend, e.g. for data analytics, should we use eigenvector of the PCA instead of linear regression?

(Generalized) linear models have a straightforward probabilistic interpretation -- E(Y|X) -- which I don't think is true of total least squares. So it's more of an engineering solution to the problem, and in statistics you'd be more likely to go for other methods such as regression calibration to deal with measurement error in the independent variables.