Yes continuity of care is one benefit of nurseries, I cm and felt awful when I went on maternity leave and my parents were forced to find alternative care, is nurseriesz. Also with a cm there is always the risk they may be ill, or one of their dc may be ill, so they may not be able to work, which would not happen in a nursery. However, if you look for cms that operate as part of a group, or a network, they will always have back-up minders, who will be known to the child through groups/activities etc, so child would not be too unsettled.
Cms under eyfs will now bw doing exactly same as nurseries, just with less children, re, 'education' and learning.
Quieter children can easilt blend into background at a nursery, its the louder more demanding ones that get more attention from my experience.
Also, of you find a cm who is accredited, you can access the free grant-funded places through them, just as you can with a nursery.
Can't advise on nanny cos know nothing !
Ooh yes, also cms will take children out and about daily, we plan all sorts of fun and eduational trips, as well as the daily things like parks, library, school run, etc, all provide lots of learning opportunities !