迎 combines walking 辶 with a figure turning to face someone: you don't welcome by waiting — you walk out to meet the guest on the road. The motion is built into the character.
欢迎光临! 'welcome, honored visit!' chimes at every shop door in China, often from a recorded voice that never tires. 迎新 welcomes the new — new students, new year — and 迎面 means 'coming right at you', head-on.
Walk 辶 out to meet them face-on — that's a real WELCOME.
The automated 欢迎光临 doorbell greeting is so universal that Chinese kids imitate it as a game — it may be the most-heard recorded phrase in the country.