The weighted dice *should* (according to Larian) work on a very short timescale. E.g., if you roll a number below 10, the next number is extra likely to be above 10. So using it should make it less likely to get multiple of the same number in a row. That's exactly why Larian implemented the weighted dice actually.
Their original rng code for dice rolling resulted in similar numbers being rolled repeatedly. If you rolled low, your next roll was likely to roll low. And the same for rolling high. Over hundreds of rolls, their code resulted in a statistically high number of...6s, 10s, and 17s iirc.
Now, it was back in patch 3 and 4 that these statistics were taken and Larian might have quietly changed their rng code (weighted and/or unweighted) by now. But assuming they haven't, it is actually more balanced to use their Weighted Dice. Getting 5 2's in a row is unlucky, but out of the thousands+ of BG3 players with thousands+ of rolls during each playthrough of EA, someone getting 5 2's in a row is practically guaranteed.
Does anyone else feel like they see natural 1s rolled with advantage more often then you'd expect?
Attempting to answer this question will result in confirmation bias. If you really want to check if 1s are rolled more often, record >300+ rolls in a row without reloading (preferably >500) and see if 1s are actually happening statistically more than 1/20th of the time. If you record the order of rolls, you might be able to identify trends too (same # rolled often in a row, bouncing between <10 and >10, etc).