Bootstrapping as a way to deal with multilevel data (similar to Bayesian hierarchical models)

Suppose you want to estimate the linear relationship between $x$ & $y$ among people. You can't afford to ask N = 100 people to measure the variables of interest and estimate the relationship, so instead you ask N = 20 people and you take repeated measurements from each individual (100 observation per person which makes a total of 2000 data points). But now your observations are not independent and are grouped with respect to individuals. You can run a linear regression for each individual separately and the relationship can be expressed as: $$y = intercept_{j} + \beta_{j} \times x$$ $$j = 1,...,20$$ So you will have 20 different relationships ( $\beta s$ ) which are individual specific but you are interested in the $\beta_{population}$ or the relationship in the population. Having a set of $\beta$ s ( $[\beta_{1}, ...,\beta_{20}]$ ) I would take bootstrap samples from them (let's say 100 bootstrapped sample of the same size) and calculate their means to estimate the distribution of the $\beta_{population}$ (assuming $\beta_{j} \sim N(\beta_{population}, \sigma_{pop}^2)$ ) with its confidence intervals. Would this approach be reasonable or its not valid in ways that I can't see?