My experience is a 5-button TRÅDFRI remote can pair EITHER to a Zigbee coordinator OR directly with an IKEA bulb. Both is simply not possible, likely as IKEA added extensions to the Zigbee protocol that HASS has been working around every since (e.g. via updates to the quirks file).
I use an automation to control Zigbee bulbs from the TRÅDFRI remote via HASS.