I think the person said that nuclear families aren't really big enough to withstand shocks is spot on. I have definitely needed more care and support from my family in the past few years, and luckily they have been there to do it. I have lovely friends, who will call up on the phone, and send me little presents and stuff like that, but fundamentally their life stage is that of being parents to small children and/or building careers, and neither of those is compatible with calling round every day to see if you need anything from the shops, or helping you put your shopping away. I am lucky as I have family willing and fit enough to help with tasks (and now retired although this wasn't the case initially), and to come at short notice- but that is partly luck, it 's also partly because I've chosen to stay near to my parents, and not seek promotion elsewhere (which is how you get promoted in my field) precisely so I can be near them.
I don't think it's about whether family or friends are 'better'- there are some truly nasty parents, but there are also some quite neglectful friends and vice versa, lovely ones of both, but in most cases it comes down to people' s time and capability- if they are 1000's of miles away from you or even 50 miles away, they can't help with daily tasks, and that's not a function of their relationship with you.
It sounds like you don't like asking for help, but your friends may not mind at all- weekends are not just for 'families' but doing all kinds of activities in my book, and we often look after kids, drive someone somewhere or fix someone else's computer on the weekend. If they said to ask for help, then do so- and you will see from their response if they genuinely are happy to do so.