I don't think the population density argument works.
Ireland (which includes care home deaths, and all deaths where Covid was present in its figures) may be less dense than England, but it's not less dense than Northern Ireland and figures are nearly double in NI than the republic. The highest death rate per capita in Ireland isn't Dublin, with by far the most densely populated areas, public transport, etc etc, it's Cavan and Monaghan, counties which border Northern Ireland.
It's very difficult to unpack causality at this point, but comparing Northern Ireland to the Republic is a pretty good example of UK policy vs another approach, and on that metric it seems the UK's approach hasn't been a success.
It's worth pointing out though there are other factors that are much harder to pull out. I heard someone being interviewed saying cultural norms were a big reason why Japan did so well, mask wearing became socially expected almost overnight and there was much less ignoring of government requests. It's hard to factor things like that in but it all plays a factor, and is similar to Swedish society in some ways. So perhaps 100% composed of a lower lever distancing policy is better than 70% compliance of a lockdown.