3 year old can’t/ won’t play by himself

I have a 3 year old son, he is an only child (we are struggling with fertility issues) he has met every milestone so far, and he’s very bright. The only thing is he seems unable to play independently, I work three days a week, so I’m at home with him a lot, as my husband works long hours.

My son does not go to nursery yet for financial reasons, he will be going in about a month or so. We don’t have any family or friends close by who have children, but he does play outside with our neighbours children very well from time to time.

I take him to soft play, and things like that where I can also, and when he is with other children he plays very well and doesn’t need me to be with him like he is at home.

I try to encourage independent play by setting him up either toys that will engage him, but he always wants to be by my side.... I get him to help me with household chores I’m doing, but I really want him to feel able to play by himself for his own good..

I feel like I’m failing in some way :( and feeling even worse that I’m unable at the moment to give him a sibling to play with, he also sneaks into our bed every night too, which we are trying to stop.

I love my son more than anything, and I just want him to have some independence from me and to feel secure enough to play for a while each day by himself.

And advise is welcome :)

Thankyou