Consequences of $\mathsf{NP}$ containing $\mathsf{BPP}$
For one, proving $BPP \subseteq NP$ would easily imply that $NEXP \neq BPP$, which already means that your proof can't relativize.

But let's look at something even weaker: $coRP \subseteq NTIME[2^{n^{o(1)}}]$. If that is true, then polynomial identity testing for arithmetic circuits is in nondeterministic subexponential time. By Impagliazzo-Kabanets'04, such an algorithm implies circuit lower bounds: either the Permanent does not have poly-size arithmetic circuits, or $NEXP \not\subset P/poly$. 

I personally don't know why it looks "out of reach" but it does seem hard to prove. Certainly some genuinely new tricks will be needed to prove it.