It comes down to the definition of LIBOR: London Interbank Offer Rate -> Every business day, a panel of large banks are asked by the BBA[*] (British Bankers Association) at what rate they would lend cash (unsecured) in a certain currency to another bank of that panel for a certain maturity, and that for a range of currencies and maturities.
e.g. Currency: USD, Maturity: 3 months -> the aggregate of the day's contributions (based on specific rules) give you the 3-month USD LIBOR fixing for that day.
Pre-crisis, as you correctly said, those interbank borrowings were considered risk-free, just as much as the overnight funding facility you could make use of at your central bank (Fedfunds in the US). There was theoretically no difference in borrowing for 3 months from another bank or borrowing for 1 month and forward 1 month in 1 month and forward 1 month in 2 months. This is what we call no basis between 1M and 3M LIBOR curves. The same was true for the relationship between all other LIBOR tenors and the overnight rate. Hence, all those rates formed a single, consistent rates curve you could use to discount risk-free cashflows.
Since the crisis, things changed resulting in two main (related) developments for the rates markets:
- Because banks realised that their peers are not risk-free, they started charging a credit spread in their LIBOR contributions which therefore created a basis between the OIS discount curve and the LIBOR curves. This means that you need to use the OIS swap curve to derive your risk-free discount factors as those differ from the IRS curve.
- Similarly, the market started to price a basis between the various LIBOR tenors of the same currency, given that lending to another bank for 3 months was more risky than for 1 month (something known as term premium).
This is what gave rise to "multicurve discounting".
Why not OIS in the first place, or what am I missing?
As explained above, the OIS was indeed part of the discount curve, but because there was no basis with LIBOR and the LIBOR swaps were most widely traded, the discounting models were fitted onto LIBOR.
P.S. Even the Overnight Index embeds a credit premium, but given that you are looking for a rate to discount say perfectly collateralised transactions, on which you would not anyway be able to call collateral more than once a day, you would be left with an inherent overnight exposure.
[*] I have not followed closely the latest developments so not sure if it is still the BBA which takes care of computing LIBOR fixings...