I found your assertion interesting, so I've looked a little further into the AstraZeneca vaccine data, the phase 3 trial results of which can be found here:
https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)32661-1/fulltext
It is in fact true that you can have the virus and then recontract it again. There appear to be several genetically distinct strains floating around which are distinct enough in spike protein epitopes to evade memory T cell response [1]. The prevalence of reinfection is even lower than the general infection rate and severity appears to be lower. However, it is also notable that the B cell antigen response drops off rapidly against even the strain you have encountered [2] and that the resilience of memory T cell populations is currently unknown, with studies suggesting that 6 months is a good estimate, and even that doesn't guarantee resistance [3][4].
This brings us on to the vaccine. As per the Stage 3 clinical trials linked above, the vaccine does work, for a while. Safety is generally fine, efficacy is fairly high, but with a caveat.
I quote the authors of the Stage 3 clinical trial, published yesterday: "In this interim analysis, we have not been able to assess duration of protection, since the first trials were initiated in April, 2020, such that all disease episodes have accrued within 6 months of the first dose being administered. Further evidence will be required to determine duration of protection and the need for additional booster doses of vaccine."
The length of the accelerated Stage 3 study was from administration of the first dose, to 14 days after administration of the booster vaccine; 73 days in total. During that time, the vaccine is effective, against the COVID strains observed by the trial cohort (and note that genetically distinct strains appear to vary geogrpahically [1]). But judging by the tailing of the antigen response, the rate of genetic drift reported and the reported and uncertain tailing of the memory T cell response, it is eminently possible that immunity will not persist for very long at all.
That might not be a problem, if we can genuinely achieve herd immunity through vaccination and eliminate the virus totally, globally. However, if vaccination is patchy or incomplete and a reservoir of virus persists anywhere, it's quite likely that you'll have a covid vaccine now, a booster a few weeks later and then boosters every 3 months for the foreseeable, possibly with variable spike protein epitopes to keep abreast of genetic drift.
Would I therefore agree with your second statement that 'The vaccine works, that's a fact.' Well yes, but with reservations.
As the authors say "Until widespread immunity halts the spread of SARS-CoV-2, physical distancing measures and novel therapies are needed to control COVID-19. In the meantime, an efficacious vaccine has the potential to have a major impact on the pandemic if used in populations at risk of severe disease."
It's a tool, not a panacea, and questions like this, is why a Stage 3 vaccination trial usually runs for a couple years, with observation out to at least 18months for persistence and robustness of immune response. 73 days really isn't that long to be immune at all...
[1]
Emergence of Drift Variants That May Affect COVID-19 Vaccine Development and Antibody Treatment - PubMed
[2]
Persistence and decay of human antibody responses to the receptor binding domain of SARS-CoV-2 spike protein in COVID-19 patients | Science Immunology
[3]
Covid-19: T cell response lasts for at least six months after infection, study shows
[4]
SARS-CoV-2 T cell immunity: Specificity, function, durability, and role in protection | Science Immunology