Is there any relation between the output voltage stability and the choice of resistors for this adjustable regulator?
The simplified calculation for \$V_{out}\$:

\$V_{out} = 1.25  (1+\frac{R2}{R1} )\$

Is only accurate when the current through \$R_1\$ and \$R_2\$ are the same.

When that's not the case (when \$I_{Adj}\$ is not much smaller than the current through \$R_1\$) then the calculation will be slightly wrong, we would need to use the formula form the datasheet:

\$V_{out} = 1.25  (1+\frac{R2}{R1} )+I_{Adj}R_2\$

Note that that formula needs \$I_{Adj}\$ as an input value!

As \$I_{Adj}\$ can vary per device (each LM317 will have a slightly different \$I_{Adj}\$) you would need to measure it for each device. That's a hassle!

If we simply make the current through \$R_1\$ and \$R_2\$ so large that the value of \$I_{Adj}\$ doesn't matter, that's much simpler!

To get that (the large current through \$R_1\$ and \$R_2\$) we need to use low values for  \$R_1\$ and \$R_2\$.

So you can use large values for \$R_1\$ and \$R_2\$ but that will make the output voltage vary more between each LM317 (unless you measure \$I_{Adj}\$ for each and adjust the values of \$R_1\$ and \$R_2\$ to compensate).

It is much easier to use low value resistors.

If that takes too much power for your use case then you should not be using an LM317. There are other voltage regulators that can work with high value resistors for the feedback network.