"with omitted doublings: popular, celebrated, television;"
words from latin or greek origin often don't double the consonant after short vowels; this is a reasonably regular pattern that can be recognised with a bit of experience.
"with needless doublings: immediately, attracted. "
not needless at all; words with prefixes ending in a consonant and roots that start with a consonant end up with doubles for exactly that reason; children are likely to be familiar with the im/in/il etc prefix (as well as others), and able to recognise roots like media and tract. It's a useful rule that applies in many words. Words like 'remedial' would make it clear that the 'media' part is a root, and words like 'impossible' would remind them that 'im' is a prefix, so shouldn't be a surprise that they are doubled.
"Other words with surplus letters:
Guide, caught, friends, temperature.
The rest contain assorted minor horrors
(laugh, design, alphabet, picture). "
the 'u' in guide isn't surplus; it serves to stop the 'g' becoming soft, and that's also a sensible rule that applies in a reasonable number of words.
the 'a' in temperature can't be heard if the word is said quickly, but again it is obvious from a study of root words, and I often encourage pupils to get into the habit of thinking of related words, as it's a useful strategy for learning unstressed vowels, swallowed syllables, etc. I'd consider 'alphabet' pretty regular by that age, as they've met lots of 'ph' words, and the 'ture' is also a common ending. 'sign' is another root that can be heard in related words such as 'signature'.
so a reasonable knowledge of phonics, and where and how certain sounds are more common in particular circumstances, would help a lot with this list, and I would expect a Year 6 child to know that.
I personally still think that there is a place for some degree of memory work with spelling irregular words, particularly the tricky bits of words, but I don't think it generally needs to be an letter by letter strategy. Just learning the bits that are somewhat irregular will help. For young children I do encourage some memory work when the words are not using common spellings of a particular sound, and they need to use those words before they have learned the more advanced spellings, but by Year 6 they should have moved beyond a really simplistic one letter-one sound phonics strategy anyway. Many words have regularities when you look at larger groups of letters. A bigger problem is just getting the children to consider those alternatives, and to think about the word and the position of the sound and to look at it and decide if it looks right, etc, rather than simply using the first choice that comes to mind, which is often too simplistic.
I think there are times when too much insistence on teaching phonics in a certain way can be difficult for some dyslexic pupils. For example, learning that 'ai' is one of several possible ways to spelling the short 'e' sound, because it does so in 'said' (and maybe 'again', but not a whole lot else), is counterproductive for my dyslexic pupils, because they are struggling enough to remember that 'ai' usually says long a. Giving them too many alternatives for each sound is overwhelming and confusing, and they then revert to rote/visual memory for everything, which is exactly the strategy that I am trying to steer them away from using most of the time. If I can get them to learn the strongest phonic correspondences solidly (which they do find quite difficult), and to then use that phonic strategy the majority of the time instead of their natural tendency of trying to visually memorise everything (which becomes too great a task and ends up leading to mixed up letter order, phonetic implausibilities, etc), then they are able to 'save' their good visual memory skills for some of the more irregular words. I entirely agree that in those words, you can work out a phonic correspondence for which letters are responsible for which sounds, but in my experience it hasn't been helpful for spelling to give so many choices to memorise for a given sound for these children. There are a few that they find easier to learn as whole words, and for others I tend to teach them the 'tricky bit' as an exception (like telling them that in 'said' you need to spell the 'e' sound with ai, rather than having them learn it as a standard correspondence; or that 'people' has the extra 'o' in, rather than teaching 'eo' as one of several combinations that says long-e, because they are then very likely to mix up 'ea' and 'eo' and forget which one normally makes the long-e sound). This is likely to be a much greater problem for those with dyslexia, however, and children who learn to read normally may take it in their stride.
I also spend time on learning where and when particular choices are more common, which includes thinking about word analysis, position of the sound in the word and other general rules (e.g., teaching 'ck' as one choice for the 'k' sound isn't helpful on its own, but rather needs to be taught as a choice that is used in a very specific set of circumstances and not in others). I dislike 'phonic' spelling lists where the words are actually quite unusual or have the sound in quite a strange position, in order to show a certain sound, instead of pointing out that that sound is most common, for example, in the middle of a word.
That list of words seems pretty reasonable to me, and fits pretty well with the sort of higher-level phonic analysis that I would want Year 6s to have done.