I haven't done a comprehensive survey, but I do know of two cycling studies that try to tackle the question of how cycling power changes with age. I do not have a background in exercise physiology or biology, but I do have a background in research methods.
Gent and Norton (2012, ungated paper here) recruited 156 men and 17 women, age 35-64 years, all active masters cyclists. They obtained 10s peak power (representing sprint power, mainly from the phosphagen or ATP-PC system) and power at VO2max (which I will call VO2max power for short) from a ramp test. Power was expressed in watts per kg. Their sample average 10s power was 15.2 W/kg (standard deviation 2.8) and the average VO2max power was 3.9 W/kg (SD 0.56). Their median amount of physical activity per week was 15 hours. This sounds like a very active sample. My average VO2max power is about 4.5 W/kg right now, which puts me well within the current Zwift requirements for the B category - I tend to be on the slower side for B races, but this is a fairly competitive category.
Gent and Norton found that the average 10s power declined by 0.138 W/kg per year, and that the average VO2max power declined by 0.007 W/kg per year (the p-values for these tests are <.0001 and 0.218 respectively). Basically, the study showed that 10s power declined faster than VO2max power; in fact, this sample's VO2max power was essentially unchanged with age. The study didn't report results by gender.
Brown, Ryan, and Brown (2007, ungated paper here) recruited 36 men and 20 women, age 16-64 years, all active competitive cyclists. They all did at least 2 high-intensity training sessions per week at the time of study. Their average amount of physical activity wasn't reported, but this was probably also a very active sample. This study tested both VO2max power and power at lactate threshold. This is basically power at a blood lactate level of 4 mmol/L, a commonly accepted threshold in the literature. I believe this can be interpreted like functional threshold power, although we determine FTP via an approximation instead of measuring your blood lactate levels during exercise. The full sample had an average VO2max power of 4.4 W/kg (SD 0.7), and a lactate threshold power of 3.0 W/kg (SD 0.6). As above, these are quite high averages. The current Zwift requirement for racing category B is maximum aerobic power at least 4.1 W/kg or estimated threshold power at least 3.36 W/kg.
Contrary to Gent and Norton, this study found that VO2max power declined by an average of 0.048 W/kg for men, although there was no apparent decrease for women. Power at lactate threshold declined by an average of 0.044 W/kg for men and 0.019 W/kg for women.
To answer the question, we would ideally recruit a cohort of individuals, and test them repeatedly over time. This would also let us estimate the individual variability in the rate of power loss - that is, it's likely that some individuals decline faster than others. However, a cohort study is expensive and it would take a long time to deliver results. Hence, we have what we have, which are two studies that recruited a batch of people of various ages at a single point in time.
It's also worth pointing out that neither study design fully tackles the problem of individuals who drop out of the sport entirely. People who decline faster are probably more likely to drop out, so the older participants in these studies could be markedly different from the average 50-60 year old. There is a quote that you should beware of an old man in a profession where men usually die young; presumably this generalizes to women and to situations not involving outright mortality. Therefore, I feel like the studies above may under-state the average decline in power by age. A cohort study could at least monitor people until they quit cycling. But they might drop out of the study for other reasons, and you can incentivize but not require continued participation.
With that said, I think it's commonly accepted among laypeople that anaerobic power should decline faster than aerobic power with age. Gent and Norton's results are consistent with that. I think there may also be physiological reasons to suspect this; for example, in older adults, the loss of fast-twitch (type II) muscle fibers is faster than slow-twitch ones.
I would have expected maximum aerobic power to decline faster than power at lactate threshold. We don't see that in the latter study.
Women live longer than men. I would expect women to decline more slowly than men. The Gent and Norton study didn't report results by sex, and they did have relatively few women. Brown et al showed that women decline slower than men, both for VO2max power and threshold power. The women in both studies have normalized VO2max power pretty close to the men, and similarly for threshold power in the second study. I suspect the difference in the general cycling population is greater, so I wonder how representative the women are of the general population of female cyclists.
What should you do
I only set out to discuss how cycling ability changes with age. I guess the prescription is try to race people roughly your age. That said, power alone doesn't necessarily determine race outcomes, and soft skills can come into play if you have enough power to stay in contention. And you can maintain significant fitness even at older ages if you keep at it. The studies didn't discuss recovery, but other answers did - you need to pay more attention to recovery as you age.
If you're interested in research methods
There's a divergence between the two studies. Furthermore, these aren't the only two out there. Readers could look through the discussion sections for other studies. Also, this needn't be limited to cycling; running can also involve anaerobic, maximal aerobic, and sub-threshold aerobic efforts depending on the discipline. My impression is that many rowing events are at shorter distances, so studies on rowers might be more informative about maximum aerobic power than sub-threshold.
Researchers would prefer to do systematic literature reviews to answer this question. Systematic means that you specify the question(s), decide on a search protocol, and then you search the literature exhaustively. That way, you aren't biasing yourself to reviewing studies that fit your point of view, although you can't avoid the bias towards published studies being different from unpublished ones. You'd then make an assessment of study quality for each question, then you'd make a judgment about where the weight of the evidence lies. In the case above, if we had only two studies about the effect of age on VO2max power, we'd say the evidence is inconsistent.
Sometimes, you can pool the results from several studies with meta-analysis or meta-regression. Some people misidentify meta-analysis as the pinnacle of research. It is not. The quality of a meta-analysis depends on the underlying literature search having been done properly. Also, you can't always pool results (for example, maybe different parameters were tested, maybe the test methods were different enough to not be comparable).