The transformerless radios were a great example of extreme cost reduction, and really they worked quite well despite the obvious shortcomings. Now over in 220V land I gather they used a resistive cord to drop the excess voltage but on 120V most got by without that. The All American Five as the design is often called was used in millions of affordable radios, bringing radio within reach of the middle class masses who may not have been able to afford the luxurious transformer powered consoles. Yes they have safety issues, but one has to remember that attitudes toward electricity and safety in general were different back then.
Not in Australia!
Almost all our 240v ac radios used power transformers (the few outliers were derisorily christened "death radios").
There were also, a
very few transformerless models made for the handful of places with DC mains ( both 240v & 110v).
The transformers for a normal mantel radio (the sort of thing which would sit on top of the fridge in thousands of households), were small enough to hold in your hand, so there was no size penalty.
With our smaller market, the cost savings for transformerless were pretty much an illusion.
Local tube manufacturers were set up to make 6.3volt heater tubes in large numbers, following the wartime buildup in capacity, & it was not worth it to setup parallel production of tubes designed for series filaments, neither was it economically viable to buy them from the USA, the UK, or Europe.
Added to this is that most series filament tubes were designed for 110v, requiring, as you noted, a resistive power cord, or other means of dropping the excess voltage, all of which were clunky, either as large as the power transformer they replaced, or not made in quantity locally.
Add to that, the requirement for additional insulation, & it's no wonder they were rejected by manufacturers in this country.
OK, radios were more expensive than in the USA, but they would have been, anyway, as they were in other countries which
did go down the transformerless route.