Kurai-
You're right, the direct restriction on gatherings is only in public places (Article 7),
However, if you have left your house to attend a bbq at someone elses house, you will have contravened elements of Article 6, so the gathering would still be illegal*.
https://www.legislation.gov.uk/uksi/2020/350/pdfs/uksi_20200350_en.pdf
edit* there is the loop hole however that argues that if you initially leave your house to go to the shop (legal), but then while out, get a whatsapp from a mate asking if you want to come round for a beer and a burger, you technically are permitted by the law to do just that, as you didn't leave your house for that reason initially, and the act of leaving your house is the only thing that is specifically restricted.
I'd suggest this is a perverse interpretation, but is an example of why poor legislative drafting can be a major headache down the line.
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Scrabble
I think the most important part of the new laws are the ones banning gatherings, yet all we've heard in the news is how the police have been enforcing the unnecessary travel part - road checkpoints in Cumbria and drones over the peaks. This (i'd guess) is the easiest to do, but has the lowest impact on preventing transmission.
The police should be focusing on gatherings.
It's Orwellian, but on Sunday, I would not be bothered if they got the drones and police helicopter out to check out people's back gardens for illicit bbqs. Easy to spot if someone has more than the direct family in attendance, and a simple job to go knock on their door and break it up. Discouraging gatherings has to be top priority IMO.