The weighted votes method seems like it would have a more negative effect on liquidity and would disproportionately reward large institutional investors that can afford to stick around regardless of the financial outlook of the company, just to hedge their bets.
Another problem I see is that weighted voting would make older shares more valuable. By purchasing old shares you reduce the number of votes it would take to do anything. This is sort of similar to having a continuous rather than discreet set of share classes. What's interesting about this is that the vast majority of non-institutional investors never vote on anything. This means that, depending on the weighting function, individuals would actually be incentivized to sell their old stock and buy young stock. This would result in consolidation of more voting power in the hands of institutional investors and founders.
The biggest benefit I see of the weighted voting method against other alternatives is a bitcoin-like FOMO buy-in in the beginning.