The 1.1 patch changed some things up a bit. The stability changes were needed but still not perfect. I much prefer the mechanics in the RogueTech mod where-in it isn't a straight up gauge that you fill up and then simply fall down, but it seems to be a check against the pilot every time they are hit with something, with negatives for previous checks (thus a gauge still) in which if you fail the check, you fall down, and you can fail the first check or succeed against a check when the gauge is completely full and stay upright (at least that is how it seems to me with my limited playing with the mod).
Feneant2, I also never seemed to get many heavies. I think I had maybe 10-12 missions in which there were heavies or larger worth getting in my playthrough (130 hours). Really, I gave up on the search for bigger, and simply searched for optimizing what I had. While I could have probably done better with some different mechs, in my entire playthrough I only lost 1 mech in 1 mission, and didn't lose a single pilot. You need to embrace what you have and simply work with it. Get multi-shot and penetrating shot on at least 2 pilots, and give them either the evasion or bulwark (then max out their tactics for the indirect fire bonuses). Get these two pilots in 2 support mechs are LRM boats with either 2x LRM15's or 2x LRM20's (if you can find something big enough, I never had a catapult or similar heavy mech, and simply used 2 shadow hawks with 2xLRM15s for almost the entire campaign (up until the last 3 missions)). You should be able to relatively easily get 2 shadow hawks as they seemed to be very common (in my playthrough at least). I forget the variant I used, but one of them can easily fit an LRM15 in the CT and one in the LT (I believe), and then put 5 tons of ammo, some jump jets (I think I had 3) and the rest as armor. With the heat changes, you shouldn't have an issue with 2xLRM15s and no extra heatsinks (the mech's engine gives you a standard 30 heat dissipation and the 2xLRMs only generate 28 heat, this even lets you jump around a bit on planets that don't have negatives to heat dissipation and still fire all weapons every round, for 20 rounds before you run out of ammo... and I didn't face a single mission that went 20 rounds of straight combat). You keep these two mechs back slightly (maybe 3-4 spaces), and always, always move their max each round (or close to it) and land in a forest (if possible).
With the nerfs you will have a little harder time as you won't knock mechs over quite as easily (I think everything got about 10-20% more stability), but it shouldn't be that hard (I had no problem knocking down at least 1 mech every turn, and once I got a heavy that could put 1xLRM15 + 1xLRM20, it became even easier). Focus on finding LRM15++ or LRM20++ which have stability damage bonus. And get them on those support mechs as soon as possible as they will really help. The idea with the multi-shot and penetrating shot skills is that you split your fire on 2 mechs when they are braced or in the woods (situational for the woods), this way you are still getting max damage ability from the mech since the penetration shot will still allow the full 4 damage on the mech in the forest or bracing. Otherwise concentrate fire on one mech/vehicle, starting with the two missile boats (once you have scouted the enemy), otherwise, use a scout (preferablly something like a wolverine which has full front armor, a couple (2-3) medium lasers and a LRM10 or SRM6 + heatsinks) to get into view range, and then have the missile boats pound the target(s), and finish up with another mech that is mostly medium laser/SRM based to get the kill (as it will typically have a free focused fire on a downed mech, and focused fire with a mech that has 4-5 medium lasers or more is typically deadly on most things that took enough general damage to get knocked over in the first place). In the beginning, refit the blackjack removing the AC's and putting on jump jets and heatsinks (you need 5 or 6 heatsinks to use those 4 medium lasers). A Vindicator is also very good with 3-4 medium lasers and a LRM10/15 (fool around with what works best for your playstyle, but ideally you want enough LRMs or SRMs to be able to do enough stability damage for knocking a non-bracing mech down either every turn or at least every other turn and to burn through the target's evasion while still doing some relatively decent damage). At some point you will get a Highlander, which becomes your kill mech, leaving you to decide if you want another bigger damage dealer/brawler (like a battlemaster as it has good speed for an assault so still scouts well, and if setup properly has enough armor to soak up the hits for being your only mech spotted for a few rounds each mission) or go for some more speed like a wolverine or black knight for scouting and rely more on their evasion to prevent damage. Don't be afraid to use the Shadow Hawks to take some damage every now and then (they do have some armor after all, and nothing in the arms, so if they get blown off, nothing lost). The idea is to present mechs to take damage so that force the enemy to spread the damage across all your mechs and not let them concentrate on one. And once you start taking lots of damage, start positioning your mechs such that they force the enemy to shoot at a side of the mech that has more armor left on it.
Use focus fire on your mechs that have lots of lasers from time to time. Putting 100+ damage on a critical area is crucial to survival. Sometimes the best idea is to neuter the enemy if you can't grantee a kill. Things like a panther with his PPC in the right arm, or look at several of the mediums that have most of their missile ammo in a single location and hit that location. The key is to mitigate the damage that they can cause, while maximizing the damage that you can cause. I have jump jets on all my mechs, as I want to be able to use the times where I can jump over the enemy and perform a focus shot on their rear center torso for a one turn kill. Again, always look for the play that minimizes the they can do to you while you still maximize your damage potential. That may mean going behind a rock that prevents 3 enemies from being able to have direct line of sight while you kill the one enemy that did have direct line of sight., etc.... Certain threats need to be killed/neutered at certain points before they can do damage. Never get within view rage of a 60ton vehicle without having firepower to kill it (there are 3 possibilities, it is a SRM carrier, a LRM carrier, or a plain old medium tank, the first 2 options are SERIOUS threats). Don't let anything with an AC20 get close to you, kill it with LRMs as fast as possible, and/or focus fire to destroy the AC20 (same principal with a mech that has 5+medium lasers). Typically I consider those the biggest threats along with LRM boats (which stock, typically don't have a ton of armor).