I think the unclarity in this question (and ironically that it's getting upvoted while the answers to it are getting downvoted) is that the question doesn't specify in its numbered premises that the side effect has to be commonly reported but only after a long time. A very rare side effect is statistically hard to find, so it's natural that it may take a long time to find some cases, even if it does show up fairly quickly in susceptible individual.
The question then makes the following blurring/confusion: it goes from a trial in which effects were (not) found to a rollout which vastly expands the population/sample. The question is then misleadingly phrased as
If not, what causes scientists to be so cautious about testing a new vaccine quickly?
There's no reluctance to test a vaccine quickly enough in a small sample (e.g. phase II trials) if the vaccine "passes the smell test" in some preclinical trials etc. But e.g. phase II trials may not find all the rare side effects. The question probably wants to ask why does it take longer to get the vaccine rolled out, but it phrases that as "scientists to be so cautious about testing". Basically the caution is about testing (with or without scare quotes) with a large sample all of a sudden, e.g. in an extreme case roll-out after successful pre-clinical trials. Increasing the sample size (as in phase III trials) gives more power to detect rarer side effects. In a nutshell
The first time a new treatment or vaccine is tested in humans, it will usually be given to a small group of healthy volunteers. [...]
The principle objectives in Phase I are to:
- make sure that the new medicine presents no major safety issues
If Phase I is successful, approval will be sought for a trial involving a larger group of people. Phase II trials will usually (but not always) include patients who have the condition the potential medicine is targeting, and aim to establish: [...]
- effectiveness in preventing the condition (if the volunteer does not already have it)
If the results from Phase II are encouraging, we will seek to start a Phase III trial. This will be a much larger trial, often involving hundreds, possibly thousands of participants coming from a range of different countries.
The principle objectives in Phase III are to:
demonstrate the safety and effectiveness of the new medicine or vaccine in the typical patient likely to use it
identify side effects or reasons why the treatment should not be given to people with the condition in question (known as ‘contraindications’)
Obviously when you increase the sample size even further (roll out) you may even find very rare side effects that were even missed in phase III. Sometimes there are so-called phase IV studies
Phase IV studies may be required by regulatory authorities or may be undertaken by the sponsoring company for competitive (finding a new market for the drug) or other reasons (for example, the drug may not have been tested for interactions with other drugs, or on certain population groups such as pregnant women, who are unlikely to subject themselves to trials).
The safety surveillance is designed to detect any rare or long-term adverse effects over a much larger patient population and longer time period than was possible during the Phase I-III clinical trials. Harmful effects discovered by Phase IV trials may result in a drug being no longer sold, or restricted to certain uses.
So the last part of the question is based on (a lot) of misphrasing and/or bad (logical) premises, lumping everything (including roll out) under the word "testing". The level of caution/reluctance is proportional with the size of the population being tested (on).
So, how long can it take to figure it out if a vaccine gives any bad side effects? How about 20 years? Because deciding if the observed side effects are caused by a vaccine or not is not actually trivial:
all three of Sabin’s OPV strains were approved for use in the US, and in 1961-62 they replaced IPV for routine immunization against poliomyelitis.
As soon as OPV was used in mass immunizations in the US, cases of vaccine-associated paralysis were described. Initially Sabin decried these findings, arguing that temporal association of paralysis with vaccine administration was not sufficient to implicate OPV. He suggested that the observed paralysis was caused by wild-type viruses, not his vaccine strains.
A breakthrough in our understanding of vaccine-associated paralysis came in the early 1980s when the recently developed DNA sequencing methods were used to determine the nucleotide sequences of the genomes of the Sabin type 3 vaccine, the neurovirulent virus from which it was derived, and a virus isolated from a child who had developed paralysis after administration of OPV. The results enumerated for the first time the mutations that distinguish the Sabin vaccine from its neurovirulent parent. More importantly, the genome sequence of the vaccine-associated isolate proved that it was derived from the Sabin vaccine and was not a wild-type poliovirus.
We now understand that every recipient of OPV excretes, within a few days, viruses that are more neurovirulent that the vaccine strains. This evolution occurs because during replication of the OPV strains in the human intestine, the viral genome undergoes mutation and recombination that eliminate the attenuating mutations that Sabin so carefully selected by passage in different hosts.
From 1961 to 1989 there were an average of 9 cases (range, 1-25 cases) of vaccine-associated paralytic poliomyelitis (VAPP) in the United States, in vaccine recipients or their contacts, or 1 VAPP case per 2.9 million doses of OPV distributed (illustrated). Given this serious side effect, the use of OPV was evaluated several times by the Institute of Medicine, the Centers for Disease Control and Prevention, and the Advisory Committee on Immunization Practices. Each time it was decided that the risks associated with the use of OPV justified the cases of VAPP. It was believed that a switch to IPV would lead to outbreaks of poliomyelitis, because: OPV was better than IPV at protecting non-immunized recipients; the need to inject IPV would lead to reduced compliance; and IPV was known to induce less protective mucosal immunity than OPV.
And yeah whether the benefits outweigh the risks of severe albeit seldom-encountered side effects is a balancing matter:
After the WHO began its poliovirus eradication initiative in 1988, the risk of poliovirus importation into the US slowly decreased until it became very difficult to justify routine use of OPV. In 1996 the Advisory Committee on Immunization Practices decided that the US would transition to IPV and by 2000 IPV had replaced OPV for the routine prevention of poliomyelitis. As a consequence VAPP has been eliminated from the US.
Yes, yes, I can see the objections already that with the current state of biology/medicine we'd figure it out faster now. YMMV, i.e. it's down to "expert opinion" whether we could completely avoid a repeat of VAPP.