Am I right in thinking that a lot of STEM PhDs students are essentially working on funded projects. So DD , with only an intercalated engineering degree, was told by the university lab she did her recent elective in, to approach them if she ever wanted to do a PhD in case they had anything suitable. No mention of needing a Masters. I have also heard it can be a bit of a tax dodge. You can give someone working towards a PhD a tax free stipend. If instead they were an employee they would be taxed so would have to be paid more.
Outside STEM it is a struggle. DS did his Masters at LSE, which we paid for. (He was surprisingly grateful. Several of his friends also stayed on but presumably had to work hard to identify funding.) He could have gone somewhere cheaper, but his Masters, Econometrics and Mathematical Economics, was known for delivering graduates with high levels of technical proficiency, something that future employers value, and which research focussed Universities are looking for. Given competition for PhD funding is intense, vanishingly few will be able to go straight from a BSc to a funded PhD. (We know of someone who did, but he consistently walked away with all the prizes throughout his school and university career.) You need a strong masters and the skills that come with it to compete.
DS applied to the US as well as the UK. (There was only funding for one PhD in DS' field at the LSE.) He was lucky to have references from some very high profile academics which helped a lot. His US funding is for six years: two years of taught Masters; three years of PhD; and a job market year. The first couple of years involved quite a lot of repetition, many US Universities don't seem to offer stand alone Econ Masters, and their UG degrees tend to be a lot less specialist than in the UK. But because of the competition he needed a Masters to compete for the PhD funding.
Out of 39 on his course, he was the only home student. The others came from everywhere: South and North American, Australia, Asia, Europe etc. Most had the same idea of a year building technical skills as the passport to technical economist jobs or academia. For many, though the fees were high, the very intense nature of the course made it a good choice and perhaps cheaper than similar courses elsewhere. Perhaps similar to a Law Conversion course. You need to have mastered the content in order to progress to the next step.