I guess one point that hasn't been made is that Riemann integral may have pedagogical value precisely because it's awkward and difficult. Look at the definition of the Riemann integral: $$ \forall \varepsilon >0 \;\exists \delta>0 \; \forall n\; \forall t_0<x_0<t_1<x_1<\dots<x_{n-1}<t_n \in \mathbb{R} \;\\\text{such that} \; a=t_0,\; b=t_n,\;\text{and}\;\forall i<n, \; |t_{i+1}-t_i|<\delta $$ followed by an inequality involving part of this data. There are more quantifiers in this expression than in any other expression that the students have seen in any of their courses by this point, plus the quantifies are over contrived sets like the set of $n$-tuples. In my experience, this is precisely what makes it the most difficult definition in the analysis course, and even reasonable students struggle with it. Usually, even the treatment of Lebesgue integral does not contain conditions that are logically that complicated - rather, the definition is split into many simple steps.

But, at some point, better earlier than later, math students have to learn how to make sense of complicated statements with many nested quantifiers. Why not when learning Riemann integral?