As a parent to a late teen now with many years experience of having other kids in the house on playmates / sleepovers etc I can say I'm really not surprised to read such lax attitudes to hygiene here.
I often had to open a window to air out the room the kids had been playing / hanging out in due to pong. Not just straight up teen b.o. but even when they were younger some of them would smell of oily, hot, unwashed hair & sour general unwashed body smell.
I saw grubby hands, filthy nails etc & blasé mums banging on about how too much washing was bad for their little darlings.
In all honesty it was often pretty grim. I've held other people's baby's that have by been washed enough & they smell like sour milk & oily doughnuts. These are the mums who don't believe in washing children. Generally middle class & into free range 'gentle ' parenting etc
We gave our dd a bath literally every night (barring illness or other v exceptional circumstsnces). We washed her long hair about 2 a week unless it needed more.
She is now 17 & showers every day & has excellent hygiene.
It's a life skill & I also disagree with those saying it's bad for their skin. We used specific products for her & she had / has perfect skin & never once had nappy rash as a baby..and I definitely think washing her properly every night was a huge factor. Along with changing her nappies far more frequently than some of my fellow mums seemed to back then.
OP I'm sorry to hear you're struggling. Life sounds hard for you & you have a lot on your plate.
I would try to up the amount of baths & build it into a bedtime routine. He's only 5 so it's early yet. It might help give some structure to the evening too.