My experience of analyzing Buffett's strategies leads me to the conclusion that he uses very broad, easy to calculate metrics to quickly evaluate a company (going deeper as he gets more interested in making an acquisition). He may not spend too much time worrying about using the best discount rates (i.e. using rates along the yield curve depending on the length of time of the discounting) but just takes a basic rate (the 10-year yield) that has at least some economic relevance, since the valuation is based on future cash flows that presumably will last at least 10 years.
I don't think the discount rate is your big problem here, though. Looking at your numbers, even with a 10% discount rate, the valuation would still be about $400 per share. You seem to have different assumptions about future stable growth, or maybe even the assumption that META will live in perpetuity than the market does. It's also possible that the FCF estimates are way off.
Even so, the 10-year rate might not be appropriate for META, since it is a higher-risk tech company. But there's not one measurement that can be used in isolation; Buffet had many metrics to find "cheap" companies than can give very different results. I cannot speak to how Buffet would value it; only what I have learned about his broad measurements.
Also, look at this quote from the page you linked:
And when Charlie and I felt subjectively that interest rates were on the low side – we'd probably be less inclined to be willing to sign up for that long-term government rate. We might add a point or two just generally. But the logic would drive you to use the long-term government rate.
Back to Buffet - his strategy was not to find stocks to passively invest in; his strategy was to look for companies that he could acquire at a steep discount and manage in a way that would improve its financials to what he thought they should be. He might use the 10-year rate (maybe plus a point based on his quote) to do a very rough valuation, look at it, and determine that the valuation is crap and ignore it, or do a different valuation that makes more sense.
It's akin to buying real estate cheaply in order to flip it, not just to buy a developed lot with the hope that it appreciates in value naturally.