I'm such a bad boy . . . but maybe you aren't pushing yourself as much as you should? If a trainer can push you harder than you push yourself, does that say something about you or the trainer? OTOH, maybe you aren't recovering as well as you could? I get crap about this every time I mention it, but I use 15g flavoered whey protein with 2T sugar immediately before and after every hard workout. I put 5g of creatine in the after beverage. I drink that one while I'm still breathing hard. I've just started this timing after reading:
Timing of amino acid-carbohydrate ingestion alters anabolic response of muscle to resistance exercise | Endocrinology and Metabolism
and it definitely makes a difference.

The Rider's Fitness book is good for what it's for, but it doesn't get you ready to move heavy iron around. For that, try:
Body By Design Book: Make Your Dream Body A Reality!
or one of the plans here:
Find A Plan

It's better if you work yourself up to it at your own pace. Start with something you can move without worry and every week increase the weight on one or more sets. You want to work up to where you will fail to be able to complete the last and heaviest set, but over a number of weeks, not right away. It takes a lot of sets to build muscular endurance up to where you can recover from hard strength workouts. The trainers are hitting you too hard, too soon.

There is a theory among trainers to build up muscular strength and endurance before introducing hard cardio. I don't know the physiology behind that, but I notice that's what they do. I do that too, I guess without theorizing about it. Every October I hit the weights at the gym and don't even start intervals until February. All that time though, I'm simultaneously bringing up my cardio hours at moderate intensity and slowly introducing periods of higher intensity.

I used to use a modified Friel plan that went like this. I did these exercises in this order:
Barbell squats
Horizontal machine rows
Back machine
Leg sled
Bench press
1-legged calf raises
Crunches
Romanian deadlift
Lat pull-downs
Start with 1 set of 30 of each with enough weight that you struggle to do 30. When you're comfortable with that weight, add a second set, circuit style, same weight. After a couple weeks, add a 3rd set. As you add the second and third sets, you can increase the weight as long as you can do all the sets with the same weight, but struggling on the last set.

Then move to doing the same exercises, but with multiple sets of each one. Start with 3 X 10 reps. Gradually increase the sets and reduce the reps, staying with about 30 total reps, until you get down to 7 sets of 4 reps. As you go, keep increasing the weights so that you fail or almost fail on the last and heaviest set. If you start in October, the 7 X 4 should be in March, so that many weeks in each phase.

I got tired of doing the above after a few years so now I've been doing the Body by Design workouts. I think they give me more balanced fitness which is nice in my old age. I'm not quite so concerned with getting the ultimate cycling performance as I once was.

Your other question was about left-right balance. Many years ago I ruptured one of my Achilles tendons. Broke it right in half. By the time it healed to where I could work that leg, it had shrunk way down. It took years to get them back the same. I mostly did it by hiking in the mountains, concentrating on stepping up with the weak leg; by doing one-legged pedaling on my rollers; by doing one-legged leg presses, and by doing one-legged calf raises. I also have a minor problem of one leg being slightly shorter than the other. That's really common. But because I cycle a lot, it has sometimes made the two legs perform slightly differently. The above exercises keep that from getting away from me. I could shim to correct, but I'm too lazy to bother with it.

One legged pedaling must be done on a trainer or rollers. In the gear you'd use for endurance work, 2 minutes with each leg at 50-55 cadence, 2 minutes same gear legs together 90 cadence, 2 minutes each leg in a very low gear, 80-85 cadence, 2 minutes legs together in the original endurance gear at 90 cadence, repeat from the beginning until you cry. The chain must stay tight all the way around the circle. Never a slack chain. If you can't do 2 minutes and hold a tight chain, do shorter intervals and gradually increase. When I could repeat these 2 minute sets continuously for 45 minutes, I was in decent shape and ready for hard work.