But Math we all want the American Dream. Who wouldn't love to be able to save and put down enough cash to buy a nice big building plot outright and build their perfect home on it? And we are entitled to, of course. But we have to get full PP first, and pay the going rate for that land, which will be probably 5 to 10 times more AT LEAST, than the cost of land with no PP. And then of course there are the restrictions on which land/locations would ever get PP in the first place.
Personally I always favour areas of outstanding natural beauty, with no passing traffic and no close neighbours, and plenty of room for my dogs to run unchecked, and room to run my business from my garden without paying business rates, but it doesn't mean I am entitled to have it just because I want it!
I live in such an area now actually, and there are fields all around me. My house is old, which is why it is here, but these days NO amount of cash in brown envelopes would allow ANYONE to get planning to build around my house. Hence it was rather expensive, and we have had to pay the full going rate for that privilege, which given that every penny we have ever earned has been taxed at source that has taken rather a long time, and we still have a big mortgage. Is it any wonder people are thoroughly pissed off when someone else pitches up at the end of the garden and takes all the benefits of that for a tiny fraction of the cost and effort? Paid for cash, from (probably) untaxed, untraceable income? Often 'earnt' through dubious illegal means?
Personally I would not want to live in such close proximity to my wider family either, but I appreciate that many do, and of course they have that right as well. But again, as others have pointed out, there are legal ways and means of doing this, and you only have to look at the British Asian community who have adopted whole towns and streets as owner-occupiers. Many of them will pool their money and live much more densely per house than the Average white British family would find palatable, but they are happy to do it, and they pay to do it the legal way, so good luck to them. I suggest if Travellers hate the confinement of brick walls they buy houses in rural areas with big back gardens and put one caravans per house out there, and stay in those as often as they flipping well like, and 'live' in their gardens! That would be perfectly legal. Oh, but hang on.......expensive. And no way of avoiding council tax. Shame. 
But what none of us have the right to do, is insist that the law need not apply to us, to enable us to do live exactly where we want, in the cheapest, quickest way possible, irrespective of the rules.
I understand what you are saying though - nothing is working so far, so we need to try another tack. But that other tack should not be just rolling over and giving in. There should be absolute zero tolerance on retrospective planning. I'd like to know (from that stuff you cited from the Guardian) how many of those legal pitches that are privately owned have planning that was obtained BEFORE occupation took place. Sometimes, I am sure, LA's do cave in, for an easy life.
And if it only takes one square mile of land to house all the illegal pitches currently causing problems (I am rather
about that but anyway...) then let's identify one suitable square mile of land. Or rather, several smaller plots scattered evenly across the country and by lottery, on brownfield sites, big enough to house a small to medium size extended family - no more than say 5 or 6 plots. Then grant planning for them and offer them up for sale as land with full PP for residential building especially for Irish Travellers or other nomadic people. But they must pay the going rate for land with full residential PP, and comply with building regs just like anyone else. And any illegal expansion of the site should be dealt with swiftly and mercilessly. No fannying around. And no special mobile/segregated Traveller services for schools. Their children enrol in school and if they don't turn up they get the authorities breathing down their necks just like you or I would, with parents imprisoned for serial absence without proof of the children being adequately educated otherwise at no cost to the state - just like you or I would be.
I hate that that may be the only solution. I hate that we should kowtow to their desire for a self-imposed apartheid system. I hate that they may need to be given the right do do this, whilst people of other ethnicities and persuasions do not have the right to express a desire for ethnic segregation on the grounds of their cultural preferences.
It sucks. But I don't know what else we can do short of totally crushing and eradicating their 'culture' altogether.