Yes, houses are routinely built on clay in Ireland, but clay needs careful foundation design. Clays can shrink and swell as their moisture content changes, particularly near trees, so foundations are usually taken deeper than on granular soils and the design must account for seasonal movement.
Clay is a competent founding material in many cases, with adequate bearing capacity for a domestic building. The complication is volume change: shrinkable clays lose moisture and shrink in dry conditions and swell again when wet. If a foundation sits within that active zone, the building can experience seasonal movement and cracking.
Trees make this worse because roots extract moisture from a wide area. Removing a mature tree before building can also cause heave as the soil rehydrates. Both situations need an engineer's assessment rather than a rule of thumb.