There does seem to be a growing issue where disabled people have to fight for the very things that only exist because disabled people fought for them. I think disabled children end up caught in the middle, but you're not unreasonable that this is a problem many deal with.
It's not helped by a lot of spaces. I've been to a few shopping centres where the lifts have those signs about considering who needs priority, and they only have a buggy sign on the sign - I always think WTF when I see those, and sadly seen those with wheelchairs and walking sticks struggling to be allowed on there too.
I'm physically disabled and there's no way I could fold down a buggy, hold a baby and shopping all at the one time and then set it all back up again by my self. So where do you go in that situation?
Other than campaigning for better transport, it's down to us to manage. I used a baby carrier with a backpack/rucksack for shopping on my back though my partne preferred a baby carrier with a backpack and/or a rolling suitcase if it was a larger shop.
As for the lady who said carry them in a sling, it’s clear she has no idea or had no children because slings stents for everybody, they’re only for the first few months- IF the mum wanted to have one. I used to have one and it’s one of the most useless things ever, but maybe that’s because I’m short. Plus what are they supposed to do after they’ve reached their destination? Have their sling on the front and their nappy bag with all the babies items like nappies and wipes on their back?
I'm aware slings and baby carriers aren't everyone's cup of tea - neither are pushchairs - but the idea that someone who has a difference experience and gives that as something they've done to get around the issue of using public transport with children must have no idea what they're talking because it's different from yours is narrow minded bullshit.
I am a shortish, disabled parent with four children, all about 2.5 years apart, and I medically can't drive so I use public transport a lot. I used a sling for all of them until they were at least 18 months old, some until they were 3 - how long they're useful and if they suit you depends on the type of sling as much as the person.
Sometimes I had to combine these with a wheelchair or mobility scooter or other mobility device and yes, I paired it with a bag on my back even when in a chair - why wouldn't I? This left my hands free for other things.
I actually found buggies painful to use and a faff to get around things in public without risking knocking into something. I didn't even own one after I got rid of one when my oldest was about 5 months until my youngest was 18 months and a floppy tantrummer and that was the awkward solution for a few months. I'm not going to say that other parents having found buggies made their lives much easier don't know what they're talking about.
When I had my first, everyone had to collapse their buggies on the route I was on, there was a tichy little luggage thing for them too, another reason I didn't bother with them. We thankfully have better now - thanks in large part to disability activists - but we still need to think of others on the bus and that includes how we manage our children. Where possibly on public transport, I think we should minimize our impact on other public transport users. Most buggie users have a lot more possibilities on how to do that compared to a wheelchair users, especially one like the OP who has already says they use a small chair.
Poor mum left stood in the cold/rain with babies/kids in tow and bags of shopping vs single adult in mobility chair or scooter waiting for next bus
These are actually overlapping groups - there are 'poor mums' who use wheelchairs with babies and kids in tow: been there, done that. Making it a competition of who is going to more miserable in the rain has no benefits.