Eurocode 7 Part 1 and and BS8004 Code of Practice for Foundations both stipulate that bearing capacity for fine grained soil should be based on both short term (undrained) and long term (drained) conditions.
In reality, when you read most text books they only consider the undrained condition. Most text books are generally assuming a normally consolidated, saturated clay. When you get away from NC clay and unsaturated clay things need a bit more assessment.
For normally to lightly over consolidated (OCR 1-2), your undrained shear strength (cu)is generally in the between 25-100kPa and as such, your ultimate bearing capacity will be governed by the undrained shear strength.
In highly over consolidated clays, when you get cu of well over 100kPa, and drained friction angles of 20-25 degrees, you drained strength will govern.
I would say you are correct in saying that most reports are based on cu only. I think this is down to the logic of that the load is applied rapidly (compared to time to dissipate pore pressure) and therefore it is an undrained loading. This is true, however OC clay dilates and creates negative pore pressure which gives an "apparent" higher strength which is lost after dissipation of pressure. This is why OC clay is scary material to work withn when assessing slope stability. A new cut may be stable for months / years, however when negative pore pressure dissipates, the slope may fail.
Having said all that, your allowable bearing pressure should be the lesser of Ult/3 and bearing pressure resulting in a settlement within your criteria (25mm is common limit, for small foundations). The "mistake" of not considering drained strength is (possibly) due to the bearing pressure for settlement is lower than your drained bearing capacity. Foundations almost never fail in shear, excessive settlement is however one version of failure.
Disclaimer : strength parameters above are indicative, there are NC clays with cu of >100Kpa, and drained phi outside of 20-25 deg for OCR clays.