Hey OP, you didn't tell us whether you have bad disks or not and that make all the difference in the world. As you get older you will get stiffer, and your core strength slips. The prolonged climbs can then really overtax and inflame your lower back muscles & ligaments. This doesn't automatically mean you have a ruptured lumbar disc necessarily, but may be simply because you are stressing the lower back musculature secondary to stiffness with aging, weight gain, deconditioning, and a lifetime of a slowly accumulating osteoarthritis of the lumbar spine, which can only be successfully treated with core strengthening. And of course the super intense the exercise, like Gary referred to in manualling, this is going to irritate the tissues the most of all.
There are many ways to increase core strength but as a sports medicine doctor, I favor resistance exercises. Yoga also works well for many people. However I want something that can be done quickly, conveniently at home, without extra cost for my patients. I have found over 47 years of practicing sports medicine, that nobody keeps up exercises indefinitely unless you meet 3 requirements: it must be cheap, convenient (in your home) and quick.
I'm presuming because you didn't mention bulging lumbar discs, and because you aren't describing severe pain radiating down the legs, that you don't have bulging discs pressing on a lumbar nerve root as the cause of your discomfort, but to be thorough, I have to emphasize that you can't definitively know this without at least a medical exam by your doctor, an x-ray of the lumbar spine and possibly an MRI, especially if the core strengthening exercises that I am going to recommend do not bring relief after 6-8 weeks. However, now that the disclaimer is in place, it is always safe to try the exercises first rather than going through the expense of medical treatment at the outset.
Resistance exercises to strengthen your core muscles means abdominal crunches with the knees drawn all the way up to the chest (easier on the lumbar discs and better isolates the abdominals), back extensions performed over a padded stool like a mechanics/doctors stool in your garage (way overpriced brand new but they last forever so buy used on craigslist or tape a small hefty square pillow to a $6.00 wooden stool from Walmart) , with the toes hooked under the side of your car so that the feet are 4 inches lower than the hips, bridges, planks held for one minute x 3 (you gradually work up to this one minute mark), and lateral leg lifts with ankle weights for the gluteus medius, possibly the most important core muscle for mountain biking, performed over the side of your bed. Additionally, you simply have to do squats because inadequate leg strength in the quads as you age will overstress the lower back on climbs as well.
You can Google these exercises to see how they are done.
3 days per week is an absolute must if you expect to see results. Personally I have found Americans are so freaking lazy this appears to be asking the impossible. 3 months max and that's it! Yes 47 years of medicine has made a cynic out of me. Now where is that goddamn tweet my patient was trying to read when I so rudely interrupted?
The old system recommended three sets of 10 reps but that takes too long for most Americans anyway. The newest recommendation is one set of five reps performed very slowly, 10 seconds up, 10 seconds down using lighter weights which helps prevent weightlifting injuries like tendinitis (except for isometric exercises like planks). This builds greater muscle strength than the three sets anyway so that is a definite plus, but it's over much quicker that's the real benefit. Squats with a significant weight are very hard to live with with this method however ha ha.
We doctors have a rather dismal success record in getting people to do what we recommend, and I have come to this very slow recommendation because most people will not continue weight training indefinitely, which you need to be an effective mountain biker, if it takes too long, which for most Americans is no longer than 15 minutes/day. And guess what? The benefit is gone within several weeks of wimping out on the exercise. Your best chance of continuing indefinitely is to make it get over very quick! One set of 5 slow is much faster than three sets of 10reps at 3 seconds up/three seconds down. Slow method equals win/win. This works for mountain bikers but not for people with big egos who want big muscles.
Finally, and I can't recommend this highly enough for mountain bikers, deadlifts using a light weight, 10-30 pounds depending on your age, your musculature and your amount of deconditioning (always begin with a very light weight and gradually progress upwards), with the above slow technique, is one of the most useful lumbar strengthening core exercises we can do. Almost immediate improvement in lower back pain can be seen on extended climbs.
However, this also presumes you are maintaining your lower back/hip flexibility by stretching your hamstrings/ piriformis/ hip flexors adequately. Tight hamstrings/hip flexors/piriformis will lead to more back pain on climbs no matter how much weight lifting you do. Stretching on a daily basis is one of those things that is so hard for people to do initially but once you have the habit it becomes second nature. Practice, practice, practice is all it takes.
Where people get into trouble with the dead lift exercise, and where it's bad reputation comes from, is when they start using heavier weights and rupture discs. Can you say CrossFit competitions?
So give these exercises a try and then go see your doctor if your discomfort persists after 6 to 12 weeks of really, honestly, genuinely doing them 3 days per week. Sorry, cynic, remember? Most likely you won't keep this up very long.
Finally, to answer your original question, yes a definite yes. Before the Ebike I was riding a custom build specialized stumpjumper that weighed 24.5 pounds in spite of heavier aftermarket components: 160 mm Pike fork, a dropper post, and 2.35 mm tires, but still suffering on the climbs with lower back pain because I am older but still very fit and an aggressive climber, with a significant old lower back injury. The pain is completely gone now that I ride the Levo (unless we do the brutal 5 hr top of the mountain ride where there are sustained incredibly steep climbs that will stress the lower back even in boost mode. However on my usual trails with a fast 90 minute training ride no back pain whatsoever. I can't recommend the E bike enough for what sounds like your problem.
One caveat however: I had to raise the front end of the bike with a 17° stem and added spacers in the steering tube to get a more upright position on the bike to completely eliminate the back pain on climbs. Proper bike fit is also a necessity.
