The characterization of the unbiased estimates as the "true" values is inaccurate, and I think this may be the origin of some of the confusion underlying the question. While it's true that the lasso coefficient estimates are biased, it's does not follow that the unbiased coefficients will correctly match the parameters of the data generation process, even if the model is correctly specified and all of the regression requirements are met. This caveat is true whether or not LASSO is used.
Additionally, the LASSO introduces its own caveats; it is an imperfect screen. The procedure could inaccurately include an irrelevant predictor or exclude a relevant predictor.
Using just the LASSO estimate, without refitting the model, implies that the researcher desires a particular trade of bias for variance. The LASSO estimates will be biased but perhaps exhibit lower variance. In the extreme case of an under-determined regression, the biased LASSO estimates admit estimation of model parameters even in the setting with more predictors than observations ($p\gg n$), but if we have reduced the model to merely $n$ predictors, the ML estimates for these parameters will have enormous variance.
That said, a few examples of similar "phased" pipelines exist. One is the "relaxed lasso", which applies lasso regression twice, once to down-select from a large group to a small group of features, and second to estimate coefficients for use in a model. This uses cross-validation at each step to choose the magnitude of the penalty. The reasoning is that in the first step, you cross-validate and will likely choose a large penalty to screen out irrelevant predictors; in the second step, you cross-validate and will likely pick a smaller penalty (and hence larger coefficients). This is mentioned briefly in Tibshirani et al., Elements of Statistical Learning (pages 91-92) with a citation to Nicolai Meinshausen ("Relaxed Lasso." Computational Statistics & Data Analysis Volume 52, Issue 1, 15 September 2007, pp 374-393). ESL also briefly mentions a LASSO-then-MLE pipeline, but does not include any particular citation.