I think you are correct that there is an aspect of "take what you can get" to the sizes that are typically used in numerical methods. Even with finite size scaling (FSS), you usually try to go to the largest size that is practical with your computational resources. Case in point: people do finite size scaling with extremely small sizes for exact diagonalization calculations.
It's worth noting that the finite size effects are themselves interesting and often contain important physical quantities
There is one more specifically-quantum feature to consider: the finite size gap and the size-temperature tradeoff. In finite-size quantum systems, even gapless quantities have a finite-size gap $\Delta \propto 1/L$ (basically because $L$ becomes the maximum wavelength). This gap contributes directly to the finite-size effects, but also separates the "finite-size zero-temperature" and "finite-temperature large-size" regimes:
- When $\beta \Delta \gg 1$, then effectively the temperature is zero, because all the excited states, even the 'gapless' ones, are suppressed.
- When $\beta \Delta \ll 1$, the finite size effects are weaker because the system 'can't see' the finite-size gap.
The computational cost of QMC (at least the form I use, SSE) roughly scales like $\beta L^d$. Because of this finite-size gap, you need $\beta \propto L$ to maintain the same `effective' temperature as you go to larger and larger sizes. As a result, zero-temperature properties are much easier to access for small sizes (with a large gap, and not-too-low $T$ requirement).
If you want to avoid finite-size effects, then you want to avoid low temperatures so you aren't detecting this finite-size gap. (Or you need a large system so $\Delta$ is small).