First-tone yān is the anatomical 'throat / pharynx': 咽喉 (throat, also a figurative strategic chokepoint), 咽炎 (pharyngitis). For the action 'to swallow' use the yàn reading.
Mouth radical on the left, the indexing radical and the body part involved. Swallowing happens at the back of the mouth where it narrows into the throat; the radical anchors 咽 in the eating family with 吃, 喝, 吞, 喉.
Right side 因 supplies the sound, yīn shifting to the three readings yàn / yān / yè. 因 originally pictured a person 大 inside a mat, suggesting 'enclosed by', a faint echo of food enclosed in the throat. Same phonetic seeds 姻 and 茵.
