You don't need T to be strong. You just need to work with heavy weights and do powerlifting style workouts. The reason is that the increased strength comes from neurological adaptations rather than increased muscle size.
This is my experience. As of last June I had not lifted any weights since 2013 and even then I only lifted for a few months. I had also been on HRT for over two years and my T was undetectable. I had gotten pretty weak. In June I started powerlifting and have worked out three times a week since then. As of today my five rep maxes in the three big lifts are:
Back Squat - 262.5 lbs. started at 130 lbs
Deadlift - 347.5 lbs. started at 225 lbs
Bench Press - 165 lbs. started at 90 lbs in Aug
All are still steadily improving by a little bit each week.
That's at a bodyweight of about 176 lbs at 5'11". I have gotten a little bigger but really not by much. Size change is marginal with a huge strength gain.
Also, in my off days when I run I often do four supersets of:
25 sit-ups
25 flutter kicks
25 push-ups
1 minute plank
And follow up the fourth superset with a five minute headstand.
While in the greater scheme that might not be all that strong it is pretty good considering I have nearly zero T, am 58 years old, and have been working out less than a year. It is also stronger than nearly anyone, men or women, who don't regularly strength train.
As for running, I used to run marathons and in 2012 had my PB with a time of 3:32. I also ran a 5K in 20 min and a half marathon in 1:35 that year. So I was fairly fast for an amateur over 50 year old. My running has slowed considerably on HRT. Part of that may be that I'm not running nearly as much as before and part may be due to my prioritizing strength training right now. But I think part is also due to reduced red blood cell count from the HRT. Even with that though I don't have any problems going out and running six miles a few times a week. I just do it a little slower than before. You just need to stay hydrated, eat well, and sleep enough.
There is no doubt that lack of T has an effect on muscle size and probably limits one's top end potential. However, you can still be stronger than the vast majority of people with consistent and focused heavy training.
If you need extra convincing, there is a female power lifter, Steffi Cohen, who has legitimately deadlifted over 500 lbs at a bodyweight of 121 lbs.
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