This claim is often made by people who lack knowledge of the clinical study process for vaccines and pharmaceuticals. Even without any knowledge of how clinical trials are designed, there are two obvious things that have been different today as compared to the past.
Technology
The first should be readily apparent. We have much better technology today to aggregate and analyze data. There have been relatively few widely used vaccines developed since the 80s, when computers started to become more available to hasten managing and analyzing data. That means that vaccine studies further back in time took much longer to analyze due to less (or no) computing power. Even with the first computers in play, it really wasn’t until the 2000s when they had enough power to quickly accomplish the type of data crunching needed for this kind of work.

Sample Size
The second obvious point that is overlooked is that any of the vaccines since that time have been for much rarer diseases, so the pools of people to be included in trials are much smaller, hence more time is needed to even have a sufficiently number of participants to statistically analyze vaccine effectiveness.
These two alone should be adequate to see why the COVID vaccine could be developed so quickly, but as a side note, mRNA vaccines have been under development for over 30 years.
The Linearity Fallacy
The last piece of the error in their argument is a little more nuanced, and some illustrations might help explain it.
Assume that one has a project that is expected to take an entire day (24 hours) to complete. This is analogous to the argument that they are trying to make, forgetting that a project doesn’t have to be done by a single person. Here’s a very simple graph to illustrate.

The black bar represents the 24 hours needed for a single person to complete the project. If two people work on it evenly, even if they start at different times, they could reduce the time to completion by 1/3 in this example. The same kind of reduction applies when splitting it evenly among three people (yellow) starting on it at different times.
The efficiency comes in when six people (green) work the same amount of time on it and start at the same time. The project is fully completed in four hours.
This is analogous to how the phases of the clinical trials for the COVID vaccines were completed. Normally. each phase of a trial is completed and thoroughly analyzed before moving on to the next one. The reason for this is that if there are safety and efficacy problems found, it doesn’t make sense to do another phase of clinical study, since these kinds of studies carry a very high financial burden for a pharmaceutical company.
During the start of the pandemic, however, there were government incentives such as Operation Warp Speed as well as financial incentives to get a safe and effective vaccine to market because there is both an ethical demand for them as well as a financial benefit due to the volumes needed.
Vaccines are generally not a major profit center for drug manufacturers. The reason is that they are a single dose of a handful of doses. The real money in pharmaceuticals is for drugs that treat chronic conditions, such as diabetes, heart disease, and others that have to be taken for years or for a lifetime. That is part of the reason for having liability protections for vaccine manufacturers in place. Without them, the lawsuits by antivax grifters and other bad actors would be such a negative incentive to innovate that most pharmaceutical industries would leave the vaccine market and focus on the high revenue chronic medications. These laws are necessary for this reason and go back decades.
Back to the sequence of phases in a drug study. There’s an excellent illustration of the difference between the normal vaccine development process and how things were done to get vaccines out for COVID in a paper by Lurie et al.

So no, the vaccines were not rushed. They were studied far more than any other vaccine prior to launch.
Safe and Effective
On a related side note, many people also seem to misunderstand what this phrase means. It’s a regulatory phrase. Safe means that in comparison to the disease being treated, the risk is much lower from the treatment than the disease. In the context of COVID, this means that the risk from the vaccine is considerably lower than that of a COVID infection.
Effective means that the treatment (in this case the vaccine) reduces the severity or risk of death of the disease (COVID). There have been plenty of studies that show this benefit of reduced hospitalization among the vaccinated, but this one stands out regarding the Omicron variant. “COVID-19-associated hospitalization rates were 10.5 times higher in unvaccinated persons and 2.5 times higher in vaccinated persons with no booster dose, respectively, compared with those who had received a booster dose. Compared with unvaccinated hospitalized persons, vaccinated hospitalized persons were more likely to be older and have more underlying medical conditions.“
The situation is similar when looking at COVID mortality when stratified by vaccination status.

