There have been parallel debates about numbers of Covid deaths across two threads so that it becomes ever more difficult to keep up with the discussion, not least since one of those thread is now on to its 15th page.
But as pointed out the number of deaths attributed to Covid in the UK, fell quite dramatically when the UK Govt decided to change the recording to deaths within 28 days of a positive test, ostensibly partly as that was the threshold that the Scottish Govt had been using.
Eurostat generally calculates deaths on the basis of those which occur within 1 year of the initiation, which approximates to the year and a day enshrined in British legal principles e.g. someone gets assaulted and if they die within 366 days then the perpetrator could be tried for murder or manslaughter.
Many deaths which are attributed to Covid may also mention something else on the death certificate. So, infected with Covid, then gets pneumonia which may be the proverbial straw that breaks the camel's back.
But if there has been no positive test for Covid then the death certificate might only mention pneumonia.
Hence, the current method used to record the stats both under-records AND over-records as would just about any method.
UNDER-records as doesn't count deaths where there has been no positive test OR (for some stats) where Covid is not mentioned on the certificate OR where any positive test or diagnosis was beyond the current thresholds used - 28 and 60 days.
OVER-records as assumes that almost all deaths within 28 days of a positive test are attributable to Covid. The exception would be eg. death in a road traffic accident within 28 days of such test unless a medical practitioner decided that being sick with Covid was causative of someone eg. not looking right and left before trying to cross the road.
...and OVER-recorded (arguably), particularly in the early months of the pandemic as accelerated the deaths of those with terminal or life threatening diseases who then acquired Covid or who did not receive the care they might have in other times.
What these stats do not count is the increased toll of death and disability that is not directly but indirectly caused by the pandemic, e.g. the person who dies from cancer due to suspension of clinics, or the inability of medics to gain access to those needing healthcare in some parts of the globe.
Similarly these stats do not count the decreased toll of death and disability from e.g. improved air quality resulting from Coronavirus restrictions - less traffic (at least for a time), ditto less pollution being emitted from factories etc. At the start of the pandemic the air quality in Wuhan changed from permanently being in the Red zone to reasonably OK.
So, arguably the most important statistics are those which measure increased (or decreased) overall mortality on a like for like year/month basis (over a sufficiently long time frame).
The last time I remember the official stats for excess mortality in the UK was before the Covid count was changed - at the time excess mortality (compared to the average for the relevant months in the previous 5 years) was significantly higher than the number of deaths attributed to Covid.
Whether comparing with the average of the previous 5 years is a sufficiently long time frame is debatable. You need to take into account variables such as economic activity (or lack of) and when e.g. the last bad flu year occurred.