How do I get my boyfriend to do more for me without overbearing him?

Okay so I've been living with my boyfriend in my family's house for about 2 years now. He has his own place but gave it up to his sister and her new family who was in bad shape and came to be with me. It got to the point my family likes him but still had cautions about him, which I understood for the first couple years of us dating (currently at 6 years) but then my family started telling me with him not around that he needs to do more but never gave me full explanations of what that meant. Now, my family says that he needs to do things for me without me having to ask him. Which I totally understand. But my issue is, for one thing I have a hard time asking others to do anything for me. I'm a very anxious person and I usually take on the work myself because I don't like relying on others too much. I get things done right away so I don't have to think about it. My boyfriend doesn't do things asap because of his job schedule, but I'm starting to see now that he takes too long to get things done. He helps me with most house essentials but my family wants him to be more of a handyman again without me having to ask him when he's available outside of work. He knows some things like some electrical and mower work, but How can I make him help me more without sounding mean?