If the Russell Group was now a fair representation of what it still claims to be

These might currently be the members (I was going to limit it to 24 as that is how many are currently in the Russell Group however, in practice, 4 universities are tied for the 24th place so 27 universities are now here). In order of research importance, based on quality plus intensity as per The Complete University Guide. I realise that this is not foolproof even as a broad way to measure it because it doesn't take in to account teaching quality so if anyone has stats on that they want to bring in to the mix, please let us know. The Russell Group was just a meeting of some universities in London. It was never anything but a biased conglomeration devoted to obtaining extra power and funding and its existence should never have had a place in a meritocratic education system.
In brackets is the number of occupational staff, not including managers, facilities, clerical staff for 2022/2023 (Source hesa.ac.uk). Generally speaking, if a university has more staff than another, you might place the university with more staff higher than my order of mention below. So, for example, Oxford might actually submit a higher number of quality pieces of research than Cambridge, purely because of the higher number of staff at Oxford:
Cambridge (5935. 5th)
Oxford (7045. 3rd)
Imperial (4390. 8th)
Lancaster (2245. 23rd) / Sheffield (3700. 12th)
Edinburgh (7695. 2nd)
UCL (9895. 1st)
Bath (1570. 29th) / Southampton (2635. 20th)
Bristol (3675. 13th) / KCL (6190. 4th)
Loughborough (1625. 28th) / St Andrews (1335. 31st)
Durham (2480. 22nd) / Exeter (3750. 11th) / LSE (1905. 26th) / Manchester (5380. 7th) / Royal Holloway (1230. 32nd)
QU Belfast (2010. 25th) / Nottingham (3610. 14th) / Newcastle (3070. 19th)
York (2615. 21st) / Sussex (2020. 24th)
Birmingham (4055. 9th) / Leeds (3875. 10th) / Aberdeen (1655. 27th) / Kent (1365. 30th)
So these might join the Russell Group:
Lancaster (2245. 23rd)
Bath (1570. 29th)
Loughborough (1625. 28th) / St Andrews (1335. 31st)
Royal Holloway (1230. 32nd)
Sussex (2020. 24th)
Aberdeen (1655. 27th) / Kent (1365. 30th)
And these might leave the Russell Group:
Queen Mary University of London (3515. 15th)
Warwick (3495. 16th)
Liverpool (3145. 18th)
Cardiff (3375. 17th) / Glasgow (5555. 6th)
As Glasgow has research quality of 86% though, which is equal with York which has less than half the staff of Glasgow, Glasgow would arguably deserve to stay in the Russell Group. It was only research intensity (57% at Glasgow v 65% at York) that threatened Glasgow here but Glasgow's high staff numbers mitigates that.
Similarly, Queen Mary University of London has research quality of 85% and over double the staff of Loughborough (83% quality), St Andrews (83% quality), or Aberdeen (76% quality) and the research intensity at QMUL is still 61%, which is only 4% less than LSE which has less than half the staff of QMUL so QMUL would arguably deserve to stay.
Liverpool arguably deserves to stay for the similar reason - its research quality is 83%.
Cardiff arguably deserves to stay for the similar reason - its research quality is 84%.
And Warwick arguably deserves to stay for the similar reason - its research quality is 86%.
So, on the grounds of staff numbers, all my suggested possible changes might not be fair to be implemented. OK but for that reason York and QU Belfast, already towards the low end in the group for quality plus intensity, might not deserve to be in the Russell Group.
Of course, it could be the case that those that might fairly be part of the Russell Group never wanted to be a part of it anyway.
So, after all that, based on general staff numbers and general research, I can personally group the Russell Group universities in the following general tiers, (in no particular order of importance, so listed alphabetically). As this doesn't take in to account factors like the quality of accommodation, personality of the place and the people, department itself, entry requirements, facilities for students, it should never be used to determine which is the 'best' place for even the cleverest of students. With no bias about assumed prestige, my results are:
S: Cambridge, Oxford
A: Edinburgh, Imperial, UCL
B: KCL
C Manchester, Sheffield
D Bristol (Lancaster would be at this level for research)
E Exeter
F Nottingham, Southampton
G Durham, LSE, Newcastle
H Birmingham, Glasgow, Leeds
I Warwick, QMUL, QU Belfast
J Cardiff, Liverpool, York

Interesting way to detail the final ranking. Some people criticise a ranking based on research (both quality and output), as they argue that a university’s ‘prestige’ or reputation should be beyond that and focus on student satisfaction, quality of students entering said uni (i.e. UCAS equivalent scores), teaching quality etc.
I align more with the uni quality linked to quality of research as it helps the academic teams to be a forefront of their academic discipline. I think the Russell Group needs to expand to include some decent unis who have the quality to be there.

Yes. As well as the names I've already mentioned, just below them are:
Stratchclyde (2010)
Heriot Watt
Brunel
Essex
Leicester (1835). Leicester is excellent for history, archeology (they helped find Richard III), medicine, modern languages and English. I have made the time to find that out as I studied English there!
Stirling
Reading
Aberystwyth
Surrey
Northumbria - splitting the research funding in the city because Russell Group member Newcastle is already there probably harms its prospects, as well as that it's 'only' an ex-polytechnic.
But the Russell Group itself seems to expect its members to keep their entry requirements on the higher side (than the more accessible grades often asked for by the above), presumably because that aids prestige when applying for funding and because it aids the likelihood of students being able to self-teach and yet end up with high grades, so the staff can do research instead.

Interesting way to detail the final ranking. Some people criticise a ranking based on research (both quality and output), as they argue that a university’s ‘prestige’ or reputation should be beyond that and focus on student satisfaction, quality of students entering said uni (i.e. UCAS equivalent scores), teaching quality etc.
I align more with the uni quality linked to quality of research as it helps the academic teams to be a forefront of their academic discipline. I think the Russell Group needs to expand to include some decent unis who have the quality to be there.

Perhaps. But there is a disconnect between this and the reason most under grads go to university. The go because they perceive that the education they will receive will enable them to get a good job. Sadly this is not the case for many and I think universities need to be honest about what under grads get.
Good researchers do not necessarily make good teachers and one has to question the lack of transparency around value added in terms of student attainment and perhaps also the proportion of students in relevant employment post degree rather than general employment.

Those are fair points. I think a good research base can be beneficial to students as they can be at the cutting-edge of developments in their fields of study.
I agree with your point about transparency. Some university courses don't deserve to exist but students are sadly taking on debts to learn subjects with no prospects.

Membership was originally based on funding from research grants and contracts not research quality.

I think that probably goes for a great many courses. I think I (teacher) and industry would like to see vocational training courses for professional careers such as computing and engineering where you cut the cr@p of going to "read" a subject, critically analyse and all that rubbish, cut out the exams (not required in the world of work) and instead focus purely on the skills and knowledge that is required for going into industry. I reckon you could cover most of an engineering degree in 1 - 2 years at a fraction the cost and actually be in with a chance of getting a real engineering job at the end of it. All a degree in engineering sadly shows is that you are good at taking exams and that is of absolutely no use to anyone.
Also - since when did an undergrads get involved in cutting edge research or the product of it? I remember all these amazing labs with fancy particle accelerators at university which were proudly shown off in the prospectuses and open days. I never went anywhere near them as an undergrad.

LSE is not in the top 30 for research grants and contracts. They’re based on subject and volume of research activity as much as quality.