Update (7/9/21): The paper was retracted on July 2, 2021. The retraction cited the same concern that not all deaths reported to national government systems are necessarily caused by the vaccine, while the paper incorrectly assumed that all such reported deaths were caused by the vaccine.
Do COVID-19 vaccines cause 2 deaths for every 3 deaths they prevent?
No. The paper is flawed and is contradicted by other reports.
Is the paper scientifically rigorous?
The paper quoted in the question is flawed. The paper searched the European equivalent of VAERS, a vaccine adverse event reporting database, compared the numbers among various countries, and then used the Dutch numbers in their analysis because the Netherlands had the most reports per vaccination. (Because of this, the authors' analysis cannot be generalized to countries beyond the Netherlands. i.e. death rates from COVID-19 may be, and in some instances are, much higher in other countries.) Additionally, the paper only tracked the number of deaths prevented for "three to four weeks" after vaccination. It's likely the vaccine will avert even more deaths from COVID-19 beyond that "three to four week" period. Therefore, the analysis in this paper is flawed. This is in addition to the fact, also noted in the other answer, that reports to these systems do not necessarily represent an event caused by the vaccine.
VAERS accepts and analyzes reports of possible health problems—also called “adverse events”—after vaccination. As an early warning system, VAERS cannot prove that a vaccine caused a problem. Specifically, a report to VAERS does not mean that a vaccine caused an adverse event. (emphasis from the original CDC source)
What do other studies say?
Luckily for us, this topic has interested the UK government. The UK government has studied the exact question of how many deaths COVID-19 vaccines have prevented. On March 25, 2021, Public Health England released a report on the benefits of COVID-19 vaccination.
Public Health England (PHE) analysis indicates that the COVID-19 vaccination programme prevented 6,100 deaths in those aged 70 and older in England up to the end of February. (Press release on Gov.UK)
As of March 14, 2021, the UK government reported 594 events with a fatal outcome after receiving a COVID-19 vaccine. (Again, reports do not indicate an event was caused by a vaccine.)
The MHRA has received 259 UK reports of suspected ADRs to the Pfizer/BioNTech vaccine in which the patient died shortly after vaccination, 326 reports for the Oxford University/AstraZeneca vaccine and 9 where the brand of vaccine was unspecified. The majority of these reports were in elderly people or people with underlying illness. Usage of the AstraZeneca has increased rapidly and as such, so has reporting of fatal events with a temporal association with vaccination however, this does not indicate a link between vaccination and the fatalities reported. Review of individual reports and patterns of reporting does not suggest the vaccine played a role in the death.
Needless to say, 6,100 deaths averted outnumbers 600 deaths reported to the UK-equivalent of VAERS by slightly over 10 times, again refuting the paper's conclusion.
Those figures have held up even to today. More recently, on May 14, Public Health England concluded the COVID-19 vaccines together prevented 14,000 deaths.
PHE estimates that 14,000 deaths have now been prevented in people aged 60 years or older in England up to 30 May 2021 (11,800 deaths in individuals aged 80 years and older, 1,800 in individuals aged 70 to 79 and 400 in individuals aged 60 to 69 years).
Estimates also indicate that the vaccination programme has prevented around 42,000 hospitalisations in those aged 65 years and over in England (approximately 5,400 admissions in those aged 65 to 74, 16,300 in those aged 75 to 84 and 20,300 in those aged 85 and over).
As of June 16, the most recent date with data available, 1,356 deaths after vaccination have been reported to the UK government. Again, deaths averted exceed deaths reported to the government by over 10 times. And the numbers above only include deaths prevented in older people.
The CDC has not yet released numbers on how many lives COVID-19 vaccines have saved. A May 6 study from The Journal of Infectious Diseases predicted that each 1% increase in coverage from 40% to 50% prevents 6,660 deaths. As of May 22, the US adult vaccine coverage is 57% (CDC source). This also questions the paper in the question as exactly 4,001 deaths have so far been reported to VAERS (see VAERS database). (Again, reports do not indicate an event was caused by a vaccine.)
Israel is a well-known case study in vaccine efficacy due to their vaccine rollout so far. Though I have not found studies of how many deaths were prevented, the number of COVID-19 cases (and, presumably, deaths) has steadily dropped with increased vaccinations (e.g. paper published by the Israel Ministry of Health in The Lancet).
Two doses of [the Pfizer vaccine] are highly effective across all age groups (≥16 years, including older adults aged ≥85 years) in preventing symptomatic and asymptomatic SARS-CoV-2 infections and COVID-19-related hospitalisations, severe disease, and death, including those caused by the B.1.1.7 SARS-CoV-2 variant. There were marked and sustained declines in SARS-CoV-2 incidence corresponding to increasing vaccine coverage. These findings suggest that COVID-19 vaccination can help to control the pandemic.