To the OP, I think the fact that you care enough to investigate your technique and to come asking for help means that you will grow into a very good surgeon. It takes time. My residency did not offer great surgical volume, and I was never allowed to break the bag during residency. My first two years, I had a lot of broken bags and other complications, and I wondered if I was a bad surgeon (even though I had been reassured by my attendings I was one of the best). Years later, I am much more confident as a surgeon and my mistakes and complications have helped me to grow as a doctor. My advice to you:
1) Patient selection and warnings - The pre-op exam is for YOU. Find anything possible that could go wrong or should be used to temper the patient expectations: high myopia, very shallow AC, guttata, irregular cornea surface, irregular topography and astigmatism, unusual AL, prior trauma or surgery, pseudoexfoliation, flomax usage, poor dilation, posterior synechiae, previous lasik, zonular laxity, ERM, AMD, etc. The ire and expectations of the patient will be much less when they expect trouble and know you are prepared for it. Use this pre-op exam knowledge to plan your surgery --> use omidria and malyugin ring or iris hooks for poor dilators and schedule them at the end of the day, have capsular tension rings and hooks standing by for pseudo-ex patients and book them end of day.
2) Investigate your surgeries and determine when and how the complication is occurring. For me, I found that I never broken the bag during cortex removal; it was always during phaco. Eventually, I realized my hydrodissections were inadequate and I was putting too much pressure to rotate lens fragments when not fully separated from the bag. Focus on trying to get a very good hydrodissection but be sure to let out a little visco before you hydrodissect to make room and decompress as you hydrodissect so you don't blow out the bag. Make sure you get a good spin before you begin your phaco. If you get resistance removing a piece of nucleus, do not force it. Instead, stop and add visco to reform your chamber and consider additional hydrodissection or using visco on a chang cannula to get behind the piece and dissect it up out of the bag. Ways you can break the bag --> phaco through the nucleus while sculpting, blow out the bag with too much hydrodissection pressure, rotate a heminucleus still attached to the bag, apply pressures to the bag from different vectors such as pulling cortex from opposite sides of the bag at the same time, grabbing the bag with your phaco tip or I&A tip, or poking through with an instrument. Focus on setting yourself up so you can't do any of these things, and you won't break the bag.
3) Do not rush or operate where you cannot see. I repeat to myself during surgery "Slow is smooth, and smooth is fast". Try not to get into a rush; it's faster not to mess up. If you can't see what you need to, then do not proceed with the surgery. Move the patient or refocus the scope. Add BSS to the surface, add visco inside the eye. Do not stick the phaco tip where you can't see. Try to keep the phaco tip in the iris plane and not too close to the wound (to avoid wound burn) but not inside the bag either. If you keep the tip where you can see it, it's much less likely to break the bag.
4) Don't make surgery hard. There is no trophy for being a hero. As I said early, if you can't get a piece out, visco-dissect it up. If you are having trouble getting the subincisional cortex, you can use the second instrument to hold the bag back or you can convert to bimanual I&A and easily reach around and get out of the cortex more safely. Alternately, sometimes I will add some provisc to inflate the bag and keep it back while i grab the cortex safely under the incision. Considering recommending femto for harder cataracts. Give yourself every tool for success. You mentioned a leathery cataract with a shell at the bottom that would not crack -- these are some of the hardest cases. Femto makes them MUCH easier to crack. Aside from this, the mi-loop could help. If not available, you keep sculpting very gently a tiny bit by bit until it will crack, being sure not to go too far through.
5) There is no one way to skin a cataract. Every doctor at my practice does phaco a little differently. I somewhat disagree with the others above that you try chopping -- chopping can be very efficient and limit zonular stress but it does recquire reaching towards the equator where you can't see as well and you really need to become good at hydrodissection and the basic steps of surgery first. Just my belief. Your technique you described is called divide and conquer. You should be familiar with this already and the basics of other types of chopping before finishing residency, but if this was not emphasized, I would recommend you go to Uday Devgan's youtube channel Cataract Coach and watch videos on different techniques. There is nothing wrong with divide and conquer, and it can be a safe way to approach most cataracts. My personal technique is to hydrodissect, sculpt a groove to create two hemispheres of the nucleus. I then crack and use my spatula to reverse chop upwards through the hemisphere, applying downwards pressure sideways with the phaco tip needle. I then use my spatula and phaco cutting tip's side sharp edge to chop into smaller pieces to emulsify, then rotate the other piece and repeat. This works for most every cataract for me. I can do this in four minutes, but typically more 5-7 minutes to be safe and to get a good IOL position, sweep the iridocorneal angle to get all the pieces and visco out, hunt under the iris for any retained fragments and make sure the wounds are sealed well.
6) Complications need not be devastating. Patients who require vitrectomies or additional retinal surgeries can see just as well. The key is managing the complications well, communicating with the patient and retina doctor. Broken bag tips --> once you recognize the tear, add visco to block any posterior flow, do not remove the phaco tip. Hook up the bimanual vitrector and make two para's and remove the remaining fragments, cortext. Perform any necessary vitrectomy. I don't add steroid to visualize until the lens and miostat are in so it doesn't fall posteriorly. Fill the bag with visco, ORA If you wish. Make sure you select an appropriate IOL 3 piece like MA60AC. Adjust IOL power for the sulcus. Widen the incision, use the bigger cartridge for the IOL, insert lens and position. Try to do optic capture if possible (lens through the rhexis posteriorly with two haptics in the sulcus) to stabilize the lens. Suture the main wound first and hydrate before removing jelly. It will keep the chamber stable and IOL and visco back. Then, using the bimanual vitrector and your two sideport incisiions, slowly remove the jelly, keep the irrigation posterior to the vitrector. You can use the irrigation tip to hold the IOL back from rushing forward. Hydrate one wound to keep chamber stability, inject miostat. Watch pupil go down and check for peaking of the pupil. Inject steroid to stain the vitreous and vitrect in the AC as needed any strands. Hydrate, check wounds for vitreous, apply BCL. Give diamox if no kidney disease to keep IOP down and load them up with steroid. I do subconj dex and q1H pred and antibiotic drops the first day or so. Check IOP next morning and make sure it stays down, look for any retained fragments in the posterior segment and refer to retina as necessary. Consider extending their post-op NSAID and steroid for another month or two longer than unusual to try to prevent CME for which they are at higher risk.
7) Don't hide your complications. I always try to be upfront with patients and show them photos to illustrate the events of their surgery, my concerns and what our plan is. Express you wish you could have given them a smoother ride with surgery but that you are experienced in managing their problems and will be sure to work closely with them to heal and achieve a good outcome. Call them to check in. I also discuss my complications and challenging cases with my colleagues to try to get their wisdom. Don't hide from surgery either -- it's important to keep your schedule as full as you can your first year or two out so you have enough cases to really practice your technique and grow. If you continue to have problems, consider recording your cases for you to review. I routinely discussed challenging cases my first year or two out with my mentor attendings from residency. You could consider hiring an experienced coach if you continue to have complications to come watch you.
Remember, we do not get paid the big bucks just for the easy surgeries. Our job is to fix the problems when the poop hits the fan. As long as you keep a level head, manage your complications and keep striving to improve you will do fine.