There are many components of physical fitness. The most ubiquitously agreed components are power, strength, cardiovascular endurance, balance, flexibility, and agility. If it is agreed that all of these are important, each should be maximized as best as possible. If there are trade-offs, the whole system should be optimized based on how desirable each component is.
Resources, however, are scarce. Unless your full-time job is to be an athlete, and you're willing to eat a lot of food and sleep a lot, one or more components will suffer. Let's assume that we want to select a single component that requires the least time for the most output and will positively affect as many other components as possible.
Flexibility – The ability to move through a range of motion at a given joint.
Increased flexibility is overblown in most people's minds. Does it really matter if I can do oversplits or put my foot behind my head? Probably not. Being able to get on and off a toilet, having sex comfortably, playing with your kids, and doing your job should be your top priorities. If you have sufficient flexibility to do these, you're probably fine. However, strength training will increase flexibility in those generally thought of as inflexible by moving through the movements under a load.
Cardiovascular endurance – The ability of body systems to gather, process, and deliver oxygen.
Although not maximally, strength training will increase cardiovascular endurance and health. During a heavy set of five, the blood pressure will increase drastically, peak, then decline and recover to normal. This is a stress that will force adaptation. It's not the same as training for a 5k or doing HIIT, but it's more than nothing.
Strength – The ability to produce a force against an external resistance
Force production is the medium by which humans interact with their environment. Force is the mechanism by which a mass' inertia is overcome and movement occurs. Because of this, increasing strength will affect many of the other attributes, because they are necessarily dependent on force production.
Power – The ability to produce a force quickly against an external resistance.
Power is force applied quickly, so by definition, power is a function of strength. It is also a function of motor unit recruitment which is primarily genetic. We can only change the things we have control over, and of the two, strength is the one we can change.
Agility – The ability to minimize transition time from one movement to another.
A function of power, agility can be thought of as power displayed quickly or with changed direction so the same reasoning applies.
Balance – The ability to control one's center of gravity.
Controlling one's center of gravity is solely a function of force and moment (force at a distance). You 'lose' balance because the deviation of your center of gravity over a balance point (fulcrum) becomes sufficiently large such that you cannot recover the center of gravity back to the balance point. Increasing strength allows you to recover from this deviation and maintain balance more effectively.
Training other fitness components is certainly important, and strength training does not preclude certain types of training, but if you want to optimize your time, strength is clearly the way to go. Strength is also so persistent that you could even choose to become strong (basically, put your other fitness components on hold), train for a year, and then go back to training the other components. Cardiovascular endurance, by contrast, is fleeting and requires constant attention to maintain goal levels.