If your baby is out and he cries, you do the same as if you're at home, location is irrelevant. I went out with baby in a sling from a young age, and frankly most people are too busy in their own lives to notice or care about a crying baby. So I did a lot of cafes, museums, shopping, at galleries, lunches with friends in those first few days, weeks and months. Then it soon became parks and soft play.
Some people love the newborn stage (I'm one of them if I'm honest). Some hate it. Some find toddlers easier, some harder. Some people find parenthood gets easier as they age, others find it gets harder or just goes up and down (I'm in that boat).
Sleep is a major factor. My first slept through at 6 weeks (11-6). Hallelujah. Then she stopped sleeping through at 4 months. She's got chronic sleep issues so sleeps 11pm-6.30am now at nine. Oh and she woke (and got us...) 3 times last night. Mostly she does sleep through though, from the last 6 months or so. But yes, it's taken 8.5 years to get her back to what she was doing at 6 weeks. Believe me, we tried everything over the years.
My youngest is 7, and for medical reasons has sleep issues. She woke 2-3 hourly as a newborn, but from 6-18 months woke every hour, often more. She's had phases of sleeping through from 2 though, but only if she's in bed with me. She's currently waking 1-2 a night, despite me being in her bed. She has prescribed melatonin to help her sleep (unusual in non ND kids, but there are reasons).
But I've also got friends whose children started sleeping 12 hours straight at 3 months without any form of training, and continued this.
Everyone's experience varies, and that's what you're doing to see on here. If someone has horrendous PND + a tricky baby, who then turned into a good sleeping toddler, then the newborn days will seem like hell. If someone has an easy/average newborn, no other issues, and their sleep deteriorated and they turn into a tricky toddler, and they are juggling work and sleep deprivation, then it makes newborn days look like a spa break. And then everything in between.
It sounds like you've got the tricky baby + pnd variety, which is why it's so tricky. You could help make it easier though, for example, you don't need to ask your mum to go to the loo. Leave baby somewhere safe, or take baby with you. Slings can mean you can just get on with your day like a human kangaroo. Also, if bf is tricky, especially with giving up dairy, then there's no shame in swapping to formula. Baby 1 above was bottle fed, baby 2 breastfed btw.
In time, even if things don't get easier, they will feel more manageable as you develop more experience in motherhood. That helps with acceptance, and rolling with the crazy ride of parenting.