In similar boat here (over 50y now, not returning to running but starting for the first time at 45).
First, you shouldn't be aiming for short distance. Their competition is generally not open unless you're in a club, and that takes more time and overhead. Aim for 5k to marathon. This is open to anyone (at least the local ones) and you can train on your own (big time saver without having to schedule/coordinate). These are also easier on old man's joints and tendons, and rely on patients and judgments (an old man strong points).
Learn the principle of good training and listen to your body so that you can improvise. The overall quantity (mileage or time), balance and variety (whole range of paces at the right dosage; varying time of the day, running surface, duration, weather) are more important than the meticulous details of a workout plan.
All that leads to: just run when the opportunity present itself. Look out for these "holes" in your life. Half-an-hour here and there (in the middle of business travel, canceled meetings, before picking up the kids, etc.). The type of training: mostly easy (I mean, real easy) first. Until you can do consistently. Use GPS watch with online account, so you have automatic logging, and can quickly see that you can get mileage up by these random schedules. Once they become a solid baseline (without causing accumulated fatigue or loss of energy at other things in life), then start sprinkling with strides and intervals (at flexible number of reps, no need to set a target in advance). Again, don't be pre-occupied by details of the workouts. Just do weekly review on how much you spend easy running, moderate and fast. The distribution should be heavily on the easy side.
Also helpful to sign up to local races few times a year, just to keep the rhythm. No need to have a particular goal of distance or time. Just enjoy the ride. I started with 10ks just to finish at the bottom 15%. Now already finishing HM in the top 30%. On to my first marathon soon.