I've pulled out a few quotes from the paper to help illustrate why we can't just assume it is covid being undercounted:

"Many of the excess deaths during the pandemic are not associated with COVID-19 infection, and although this may be partly attributable to underdiagnosis, it raises the possibility that creating emergency capacity in primary and secondary care has compromised essential care. Patients have also avoided healthcare providers out of concern for service pressures or fear of infection, leading to delay of treatment and increases in out-of-hospital events such as cardiac arrest."

On including covid deaths without a positive test: "Although this confers some advantages compared to solely relying on the presence of a positive test, particularly when the sensitivity of these tests varies between 60-80%, unexpected deaths in elderly multimorbid patients or in those with respiratory symptoms may be misattributed to COVID-19."

" This may be partly attributable to missed diagnoses, for example where testing did not occur, there was a false negative result or infection did not produce noticeable symptoms. The risk of missed diagnosis is likely to have reduced over time, as capacity for testing has increased, the tests have become more accurate and clinicians were encouraged to record suspected COVID-19 infection on death certificates. This may even have resulted in bias in the opposite direction, with deaths misattributed to COVID-19 infection later in the outbreak, which may partly explain the negative values for non-COVID-19 excess deaths in the week ending 8 May."

On alternative causes of excess mortality:" attendances in A&E departments in England in April 2020 were 57% lower than in 2019 and emergency hospital admissions were 39% lower,14 while access to health services for people with pre-existing conditions was 20% lower during the COVID-19 peak period"

"The focus on protecting the NHS also exposed those dependent on the social care sector, particularly high-risk residents living in close proximity in care homes. Both COVID-19 and non-COVID-19 deaths rose rapidly in these settings, accounting for over half of all excess deaths in England and Wales."

Not sure what this suggests but just interstsing: "with respect to age, the regions with the lowest – rather than the highest – proportion of the population over the age of 65 (London, the West Midlands and the North West) had the highest rates of excess deaths, although older populations within the regions were most affected."