Isn't it the case that with low (even if stable) prices, any large transaction in an exchange will move the market, thus causing volatility? The recurring joke here is that current btc economy cannot even sustain a fancy yacht or a house being traded without volatility.
The only way to a more stable and functional market is increase in "market cap" (a misnomer) - and since supply is set in stone, this means increase in exchange rates (influx of fiat capital).
Yes. However two things are needed. A larger money supply AND sufficient market depth (doesn't matter where it will all arbitrage). For Bitcoin, the larger money supply requires higher exchange rates (not just pump and dump but sustainable higher exchange rates due to real demand). The deeper market depth will come with higher prices, larger players, more economic activity, and more mature exchanges/platforms.
The USD:EUR exchange rate has low volatility not just because both economies are large but also because the Forex market has massive depth. It takes hundreds of millions in trades to move the market in any meaningful way.