@OP:
Bitcoin owners/traders/companies interested in stabilizing the bitcoin price could "unite" into some kind of special pool of traders.
This has been proposed as early as 2013 here at Bitcointalk (it was in March/April, just before the $100 mark, the first time I read about that). I don't remember the name of the exact thread, but the idea was the same: coordinate a number of traders to stabilize the price using "contrarian" trading techniques, at least to correct the most blatant under- and overvaluations. Basically "buying the dips" and "selling the peaks".
The problem is: someone will try to find a vector to profit and attack this scheme. There was a stablecoin called NuBits in 2014 or 2015. It tried essentially what you're proposing, coordinating and incentivizing traders to stabilize the currency around a USD 1 peg, but failed miserably after some months due to an attack on the peg. (Yep, and we all saw what happened to Terra/Luna these days
).
You can now say that the goal is not a peg but simply a smoother curve. The problem here: There are real variations in adoption/the supply-demand curve through the months and years. There are phases when the interest in Bitcoin grows, and so also demand grows. When these phases come to an end, supply (on exchanges! not the blockchain ... the 21 million limit can obviously not change) grows and a downtrend is likely. And vice versa.
If your trading group now tried to keep price relatively stable, e.g. inside some sort of "price corridor", there would be simply situations where the "corridor" would be unsustainable because either demand or supply are too high. The price would then very likely crash hard or FOMO hard, and the traders of the group who stick too closely to the "stabilization rules" or don't adapt would go broke.
I somewhat agree also with those arguing that you shouldn't try to stabilize against a potentially unstable fiat currency. (A basket of goods and services would be better.)
Now: what could instead to be done? I've thought for years about that. There are different possible concepts, no one is perfect but all have different advantages:
1)
A decentralized system to back Bitcoin with goods and services. This could work in the following way: Any merchant who sells goods or services would be able to participate (e.g. with a plugin for their ecommerce CMS, or on OpenBazaar). They put a price guarantee in BTC for i.e. 24 hours for their products. If the price of BTC tanks, then the customers are able to get the goods/services for lower prices. But: the merchants can also profit, as in downtrends they would see their sales volume increases as customers get the goods for a "discount". (This is above all interesting for those selling digital goods as they have very low or inexistent - if they're sold from the producer - unit costs.). Important: that doesn't need centralized coordination at all, each merchant independently sets their own rules.
2) Using financial products (as smart contracts or on centralized platforms) to follow
Moving Averages, preferrently longer ones like the EMA-200. While this would not affect Bitcoin's stability directly it would give an opportunity to risk-averse holders. They could invest in a product which represents a SMA or EMA, and thus be somewhat protected from the worst crashes - but you trade it of course for the possibility to sell at bubble peaks.
3) Using options (e.g. put options to protect against a too deep crash) and techniques like collars. Could potentially also be done
in a decentralized way using atomic swaps.
4) In general: use Bitcoin more as a currency, as this leads to more liquidity and also to create price expectations (e.g. "A house is worth around 10-20 BTC").
All these things would not impact Bitcoin's stability directly, but indirectly: Risk-averse groups would have methods to protect themselves from a crash while profiting from the general positive longterm trend. So they would be more unlikely to panic in a downtrend, and the price curve could get smoother. They also would potentially increase liquidity (goods and services are also a form of liquidity) and thus lower volatility systematically.