Some vaccinations need only be given once and my (non-medical) understanding is, despite that, they continue to provide you with protection as the antibodies/t-cells they produce circulate in your bloodstream/lymph system. Others need to be topped up after a given length of time but their purpose is the same, to generate antibodies and t-cells to keep you and those around you safe (or at least safer).
The funding of research is worth thinking about. Most vaccines and medecines take years to create and test. But that, to a large degree, is down to funding. You get an idea and then look for funding which might take you part the way to your goal. But then you need to find more funding to move you along and again and again. The size of your team is dependent on your funding.
In the case of Covid it happens that much of the research had already been done. The underlying technology of the AZ vaccine has been around for years. What they had to do was identify the virus and work out how to insert that into the pre-existing technology. Instead of having to look for funding they had it thrown at them; they were able to recruit much larger teams (but, yes, they wern't gigantic, they didn't need to be) They could start producing test batches without waiting for more funding. Instead of recruiting small batches of test paitients they had people lining up to be guinea pigs.
All this meant accellerated development and testing.
What about long term safety? When a new vaccine is approved the usual test period will have been longer than for Covid because of the funding issues but I go back to the fact that the underlying technology of vaccines is well understood and tested. When any new vaccine appears the number of people it will have been tested on will be miniscule compared to the numbers that might be given it so we learn as we go - just like the Covid vaccines.Yes, there were some health scares along the way but look back at the history of vaccines and you'll find that they all have an element of risk but I would suggest that most of us have had vaccines over the years without a 2nd thought - I've lost count of the number of vaccinations I've had.
This book is worth a read (written by the AZ team)