The general advice is that we should aim at 90 for an average cadence, and pushing slower than that can produce knee pain and injuries, and back pain and injuries.
However, everyone has their own best cadence. For example, I have not raced and find 110rpm is good for me on a flat-ish ride. My brother who has raced stays at around 120rpm on the same terrain. I find I'm changing gears if my cadence drops to 100, unless I'm on a long hill greater than about 6%, when I drop below 100rpm with my 39x26 lowest gear.
Climbing ability is a combination of your strength, endurance, and the gearing. Cadence enables the delivery of higher power. As one example, look at runners - sprinters aim for high step rates. For a cycling example, look at sprinters on the track or at the end of a road race. They also aim for high cadence. Going back to my personal examples, I can push to 180rpm in a sprint, and my brother passes 200rpm. Cadence is where the power is. If the cadence is too low on a hill, then we need better gearing to get back to where the power is.
Cadence is also where the muscular endurance is. For a given speed, at 50rpm your legs are pushing twice as hard as my legs at 100rpm (all else being equal). So my legs are generating half the power per revolution than yours. Cutting a long story short, the legs working harder will tire more quickly.
Cadence is supported by your cardiovascular capacity. A good cadence is where your performance is within your Aerobic capacity. This is essential for endurance.
So, to answer your headline question -no, increased cadence should not hurt your climbing strength.
The reason for the caveat is your mention of interval training. Interval training is about increasing your power and lactate threshold, not cadence. It does this by going faster for short periods, requiring you to work at or above your aerobic capacity. While valuable for developing endurance, it is not about training for cadence. Cadence training requires pushing a light load (for example rollers or riding behind or scooter) for long periods.
BTW, using the calculations provided by R Chung in this answer, suggest that your sprint on a 12% hill is producing close to 2kW. If your figures are correct, then that suggests you should be racing :-)