I think Leeds and a few other of the original civic Redbricks are easily on a par with Bristol, Durham and (especially) Warwick (I don't know why this is always thrown in with those, it's more like York).
The original civic Redbricks are arguably even better than some of those because of their funding/enormous endowments, their historical reputation (this is MUCH more important than a university's 'ranking' in my view, not so much the university's age but whether or not it's consistently had an excellent reputation throughout much of its history, i.e. before league tables were devised), the number of Nobel Prize winners associated with those universities (not so much on based on temporary staff but amongst people who actually studied there), the level of contribution towards the academic community, and (to a lesser extent) the notable alumni. These things generally feed in to how 'respected' a university is, not where it stands on something as fluid and as changeable as league tables (often based on some very questionable criteria).
The only reason places like Leeds aren't quite as reputed as those universities on TSR (and it is mainly TSR) is because they aren't targets or semi-targets for IB, and because their average entry tariffs aren't quite as high. In the real world, most people don't care whether or not your university is strong for Maths and Economics, so for most subjects and careers this has no relevance. Also, the entry tariffs aren't as high because they tend to be bigger universities with big nursing schools, more foundation years and access schemes, which lowers the average. If you ignore those things, you'll find the average entry tariffs and the volume of people with top grades at all these universities is similar (there should be another measure for this such as "average entry tariff compared to student population size" to get a more accurate picture). Just because a university is smaller, that doesn't mean it's "better". Graduates aren't penalised for choosing to spend 3 years in Sheffield over Durham.