A very subtle question, I like it!
Well, there's an implicit long bias anyway, because most markets have gone up over time I will always have more +20 and -20. And the effect you describe is probably strongest in clear risky assets like equities (or risk off assets like vol).
There is some analysis that goes part of the way to answering your question in these two posts:
https://qoppac.blogspot.com/2020/07/do-non-binary-forecasts-work.html
https://qoppac.blogspot.com/2020/09/forecast-linearity-and-forecasting-mean.html
So for example, the plots of forecast vs realised return show that if anything the forecasts aren't long biased enough (a zero forecast associated with a positive ex post return). And the second post talks about making the vol measure more long term in nature - something I've implemented - but not for the reasons you state.
... but I haven't actually explored this specific topic. My gut feeling is it will make a very small difference, and since it only works on a subset of assets I'd be reluctant to implement.
Still, as they all say, "Further research is needed". This question might even make it into my next book (I will credit you)
In fact I do web scrape from the website you name, but only for historic data.
(I paid them for a premium subscription as well - did some multi year deal which finishes in 2023. I won't be renewing, as I will probably have added all the markets I need by then)
I've considered web scraping, but it's extremely hacky to get right, and you're at the mercy of the provider if they decide to change their formatting or website in one tiny way, or worst still drop the data entirely (like yahoo did). That's fine for historic data where the process of backfilling is quite manual anyway, but not for production data.
We've had a debate on this thread before about whether it would be worth getting daily data for an ICE market, and then trading 'blind' without live market data. And with my new dynamic optimisation I would also have the option of bringing in data, calculating an optimal position, and then not trading the market (but that market would give me useful information for a market I could trade). But incorporating another data provider into my production pipeline, especially with the potential it could go wrong, is a big faff and probably not worth the very small marginal benefit (although it would be nice in particular to have short sterling and euribor in my portfolio, for sentimental reasons).
GAT