As @thosphor pointed out in comments, the very existence of the sport of triathlon proves that it's possible to be good in up to three endurance sports simultaneously. Thus, with respect to the OP, the question in the title is trivial.
Should we try to be good at more than one sport?
This may be a more interesting question, and some of the OP's phrasing implies that this may be the question they were getting at. I'm going to answer based on what I know of physiological and sports training principles, but I am not a specialist in the subject and I would welcome corrections from someone who actually knows the science.
When you start a new sport, part of the initial physiological response involves your brain learning to recruit the muscles involved. Thus, when you first lift weights, you don't actually gain much muscle mass, but you do gain strength as your brain learns how to fire your muscles. I would expect that all sports involve some sport-specific fitness as well as generic fitness. This corresponds to the sensation of fatigue that the OP reported after taking some time off for running.
The OP identified running as the chosen supplementary sport. Running is an endurance sport like cycling. Thus, once you achieve running-specific fitness, you would merely be adding to your general endurance. For that reason, it wasn't clear to me that you would expect running to improve your cycling fitness, provided you are getting sufficient overall training stimulus to achieve your goals. I have not digested the source that DSway provided in his answer yet, and I note that running has some logistical benefits (can exercise when away, can get the same training stimulus in less time).
Offsetting that is that you will need some minimum amount of training before you can run far enough to get enough of a supplemental training effect. Presumably, former runners will need to train less than complete novices, but even former runners who have not run in some time will have lost their running-specific conditioning. I can attest to this from personal experience.
Children should not be forced to specialize
Admittedly, this doesn't have anything to do with the OP's direct question, but I argue it's a reasonable tangent, and it is related to the question if we should diversify our sports.
Ross Tucker, a sports scientist who writes for the blog the Science of Sport, argued that forcing children to specialize early is counterproductive. Kids should instead be allowed to sample various sports, and to decide if they want to specialize at their own pace. If anything they should be pushed to delay specialization. Early specialization seems to be associated with both injuries and burnout. So, one might actually want to push a child to try out other sports.
The goal of his statement is to avoid the latter two outcomes, not necessarily to promote diversification for its own sake. Extrapolating from that to adult cyclists, I would argue that we should sample some other sports to see if there are any we like. I think we should not feel obliged to pick up running, or any particular sport, if we don't like that sport.
One potential exception is that sometimes, multi-sport training can help address overuse injuries. For example, I know that I benefit from weight training, as I would otherwise have more back and shoulder pain. For someone who runs very long distances, I wonder if they would benefit from diversifying into a non-weight bearing endurance sport (presuming that they enjoy it!) to reduce overuse injuries from running. Because cycling is not weight-bearing, perhaps those of us who are at risk for low bone density should try to find a second sport that is weight-bearing, like running or weight training.
As a side note, some sports do inevitably require early specialization to achieve an elite level. I believe these would tend to be sports that require a very high level of technical prowess, like gymnastics. These would be exceptions to the header. Related to this, swimming is very technique-intensive, and I suspect that most triathletes who do not have early swimming experience will have the swim as their weak sport.
Can athletes achieve a professional or elite level in multiple sports?
This is another question that may be related to the title, although I assume the OP may not find this personally relevant. Can an athlete achieve a high performance in multiple sports? Related but not identical, we might ask if it's possible to be world class in more than one sport, or world class in one sport and to play at a very high level in another. For example, I recall that Michael Jordan was one of the greatest basketball players and that he turned professional in baseball, even if he didn't come anywhere near the same level.
In addition to triathlon, decathlon and related multiple track and field sports, biathlon (shooting and skiing, derived from alpine warfare), and Crossfit all attempt to build proficiency in multiple related sports. So, clearly it is possible to perform at a high level in multiple related sports, e.g. different endurance sports, different ball sports, different combat sports, different strength or speed events. Some comments below noted athletes performing at high levels across multiple similar sports. In addition, Mavi Garcia came (late!) to professional road cycling from a duathlon (run + bike) background. She nearly won the 2020 Strade Bianchi race, beaten by Annemiek Van Vleuten, who is the dominant women's road rider of the moment.
However, we do not regularly hear of multi-sport athletes being in contention for world class status in related single sports or single track and field events. By world class, I mean being in regular contention for a podium spot in multiple high-prestige races in a chosen sport. I suspect that each individual sport requires a great deal of sport-specific fitness and technique and other forms of specialization. In cycling, the sport-specific techniques and skills include racecraft (e.g. which line to take in a hectic race, how to identify a break that must be covered, which rider to get behind for a sprint, etc) and bike handling. Those posters here who have also tried running can probably attest to the fact that they probably had different muscles being sore after a run than after their usual rides - this is an example of sport-specific fitness, even between two endurance sports that use the legs for propulsion.
I would suspect that to achieve world class status in one sport requires an immense amount of training in that sport. I would wager that most athletes don't have the time in the day or even during their entire careers to achieve world class status across multiple sports. If this were possible, we should more regularly hear of, for example, world class decathletes winning prestigious 1500m races on the track or Crossfit athletes transitioning to powerlifting, or vice versa. Perhaps we shall do so as time goes on and as multisport events gain more prestige.
With technique-based sports like ball sports or combat sports, there is the additional complication that the techniques and rules are very different, and there is a lot of muscle memory that one would have to un-learn. Most readers probably just remember that Michael Jordan didn't make the truly elite level in baseball, but that misses the point that they are very different games. This makes his transition doubly impressive.
It seems intuitively obvious that attaining a high level in different types of sports, e.g. strength and endurance sports, is impossible. Coffey and Hawley write that the metabolic pathways used to adapt to aerobic training will inhibit the pathways used to adapt to resistance training, and vice versa. Thus, it should be impossible for a world-class marathoner to simultaneously be competitive in 100-400m sprints, let alone to be world class at those shorter distances. This is not to say that endurance athletes should not work on strength exercises to enable their endurance pursuits and to enhance general fitness; it merely means that endurance athletes will not be able to attain their optimal strength if they were specializing in strength sports.