Good question. I suspect this was because when the rate was defined in the 1980s, collecting that data would have been difficult.
After the LIBOR scandal, the EU brought the benchmarks regulation (BMR) which says that interest rate indexes have to be based on actual transactions, just as you say. Euribor, the equivalent of LIBOR for lending in euros, was reformed to be based on transactions:
But in the end, US and UK regulators decided just to abolish it, in favour of overnight indexes based on real transactions (SONIA for pounds, which already existed, and SOFR for dollars, which was created for this purpose).
I believe this divergence happened because of differences in the lending markets. In the euro area, there is still a lot of unsecured term lending, which is what Euribor measures. But in the UK and US, this kind of lending has largely dried up, but there is a lot of overnight lending, so they chose rates which measure that. I don't know why the euro area is different to the US and UK here. It's possible that the euro market will evolve to be more like the US and UK, in which case Euribor will stop being credible, and the euro will also move over to its overnight rate, ESTR.
Another fun quirk is that SONIA and ESTR measure unsecured overnight lending, whereas SOFR measures "repo", which is essentially lending secured with government bonds as collateral. There is a sterling overnight repo rate, RONIA, but i don't think it's used much. I think repo volumes are higher than unsecured lending volumes; if that difference gets stark enough, perhaps sterling and euro regulators will force another switch, to the repo indexes.
After the LIBOR scandal, the EU brought the benchmarks regulation (BMR) which says that interest rate indexes have to be based on actual transactions, just as you say. Euribor, the equivalent of LIBOR for lending in euros, was reformed to be based on transactions:
https://www.emmi-benchmarks.eu/benchmarks/euribor/reforms/
The administrator of LIBOR proposed doing the same:
https://www.clarusft.com/rfrs-libor-is-changing/
But in the end, US and UK regulators decided just to abolish it, in favour of overnight indexes based on real transactions (SONIA for pounds, which already existed, and SOFR for dollars, which was created for this purpose).
I believe this divergence happened because of differences in the lending markets. In the euro area, there is still a lot of unsecured term lending, which is what Euribor measures. But in the UK and US, this kind of lending has largely dried up, but there is a lot of overnight lending, so they chose rates which measure that. I don't know why the euro area is different to the US and UK here. It's possible that the euro market will evolve to be more like the US and UK, in which case Euribor will stop being credible, and the euro will also move over to its overnight rate, ESTR.
Another fun quirk is that SONIA and ESTR measure unsecured overnight lending, whereas SOFR measures "repo", which is essentially lending secured with government bonds as collateral. There is a sterling overnight repo rate, RONIA, but i don't think it's used much. I think repo volumes are higher than unsecured lending volumes; if that difference gets stark enough, perhaps sterling and euro regulators will force another switch, to the repo indexes.