Separation anxiety

I'm a FTM and everyone keeps telling me my son's separation anxiety is because I'm always with him (SAHM with zero family around) I think that's false, I believe it's a phase every child goes through but I'm curious what everyone does for it? My "alone time" is grocery shopping on Sundays and I leave when him and my husband are playing or doing something. Sometimes are harder for him but he'll settle down. My mil keeps telling my husband to make me leave him more but I don't see the need to start something new while his anxiety has been at the highest. I feel like if he knows I'm not going anywhere and if I do, I come right back he might overcome it sooner and not have separation anxiety his whole life. Also my husband works 6 days a week so more alone time isn't something that would really work for us. What is your take on it?