Disclaimer: I don't play online, only vs. AI on close to max difficulty settings.
Let's address the changes one at a time.
Sensor Lock:
Old: Always kind of a meh ability that was sometimes useful when you didn't otherwise have a shot, or felt the need to brace, and occasionally getting in some long range LRM fire.
Beta: A definite, if only minor improvement, though it does make for some interesting decision making, especially when coupled with PPCs. I would say that it is a good change that doesn't feel overpowered, though it also seems like it could be, if one was inclined to go for the full on 'control the battlefield' mode.
Bulwark:
Old: Apparently the community-at-large's go to skill. It allows your mechs to face-tank a considerable amount of damage, though I don't think it is as powerful as many people seem swear by, as most maps have a fair amount of cover, which absorbs a reasonable though less amount of damage.
Beta: Being nigh on invincible is cool, I guess, though while your invincible, you're also not shooting, as you have to brace in order to get it. If your not shooting, most of the time, you're probably doing it wrong. I have more to say about this below.
Juggernaut/Coolant Vent:
Old: Juggernaut was somewhat useful early in the campaign but gets worse the further along you go. I think it has 3 main problems. First, the farther into the game you get, the heavier the mechs get, which means they get less mobile, making it more difficult to run over and bash that guy over the head. Second, there is no 0 Initiative phase (there really should be a 0, come on Devs), which causes it's ability to essentially do nothing in the late game. Thirdly, it's in a skill tree whose first ability wants you to stay still, conflicting interests to say the least.
Beta: Coolant Vent is busted, as others have pointed out. Cockpit Mods negating the drawback should not be a thing, also, fluff text flavor fail, though I get what they were meaning. Assuming the Cockpit Mod issue gets resolved, I think this is probably to powerful, though I need to playtest some more with it. It lets you build an extremely high heat Alpha Strike mech that you then get to Alpha Strike turn after turn after ludicrous turn. Late game, with a fully upgraded MedBay on the Argo, the downtime from injuries is not the a big a deal, so deliberately injuring your pilots for a potential turn of devastation is a small price to pay. To borrow/paraphrase a term/concept from MTG, your pilots Hit Points are a resource, use them. I see this as an ability that is likely very abusable, and should be drastically altered or ditched altogether.
Evasive Movement/Sure Footing
Old: This in my opinion is the better defensive skill. I must disagree with the Devs when they said It was so rarely relevant. Late game, on Assaults and Heavies, and even on some mediums earlier on, like the Hunchback, it is VERY relevant. That extra pip meant the difference between getting hit and not getting hit.
Beta: Surefooting is worse than Evasive Movement, as you no longer get the extra pip if you jump/sprint/melee/DFA, and the auto entrenching in no way makes up for the loss of evasion. More to say below.
As a member of the 'evasion is superior to bulwark' (in most situations) camp, I found that jumping back and forth in cover gave as good or better survivability than face-tanking with Bulwark, and obviously, you were much more mobile and maneuverable. That being said, I think the change to Bulwark was too drastic...feeling more like a ban-hammer than a nerf-bat; it feels basically useless and unplayable. This might be due to my playstyle, which is very mobility oriented, LITERALLY putting jump jets on everything (don't leave home without them), and never bringing more than one mech with Bulwark (1.2 version), and the Bulwark 'mech is always some kind of fire support build, tons of LRMs/PPCs/AC10s. Just placed him on a hill with good fire lanes and proceed to rain death, but I digress, my apologies...
The change to Evasive Movement is, in my opinion, a worse nerf, as that extra pip while jumping has seemed very crucial on numerous occasions, especially late game with my Assault Lance going up against 7-8 OpFor Assaults; I very often had 4-5 pips of evasion on my Assaults, by constantly max jumping back on forth in cover/LOS blocking terrain. Without the evasion, and the nerf to Bulwark, two lances of OpFor Assaults seems daunting/impossible (pardon the hyperbole) on the higher difficulty settings, and a challenge even on normal.
All that being said, I don't think either of these (non-Beta changed) are even the most powerful/must have ability on all your pilots. That, in my opinion, would be Master Tactician, followed very closely by Multi-Target. Every single one of my pilots is either Master Tactician paired with Bulwark/Evasive Movement/Multi-Target, or Multi-Target+Breaching Shot paired with Evasive Movement/Bulwark, with the Master Tactician making up ~80% of my Pilots. Please Devs, don't nerf Master Tactician, but if you were going to nerf one of the abilities, it should've been that one.
I think this more a general game system problem, not just BattleTech, but any game that utilizes Initiative as a core gameplay mechanic; anything (mostly) that modifies initiative is BUSTED, whether it is a buff to yours, or, to a lesser extent, a debuff to your opponent's. These sorts of abilities allow a player a degree a battlefield manipulation that perhaps they shouldn't have. In BattleTech, I manipulate it by finding the OpFor on Sensors, then maneuvering as close to them as I can without actually getting a visual, all while maxing out (or nearly) my evasion, preferably in cover/jumping+brace. I then proceed to Reserve my Lance through to the last Initiative step, letting the enemy come to me, dodge most of their fire, fire upon them, stripping their evasion, then in the following round, due to Master Tactician on all/most of my guys, get to activate first, thus getting to get all my evasion back, while tacking advantage of their lack of evasion, due to being shot up on the initial encounter in the previous round, and proceed to murder them. Rinse Repeat.
I think a better way to address the issue of everyone's Pilots having the same abilities would be to limit the number of Pilots with the same ability to be deployed on the same mission. One Bulwark, One Master Tactician, etc. I also think that the skill trees are part of the problem, for two reasons. First, end of the tree abilities ought to not only be powerful, but FEEL powerful, currently, only Master Tactician is like that (at least for me). Secondly, I don't think 4 skill trees with a total of 8 abilities is enough in all honesty; the Devs created overlap where there shouldn't have been; Bulwark meet Juggernaut, Juggernaut, meet Bulwark, wait, what?! Personally, I would double the number of trees(8), quadruple the number of abilities(4/tree), and double the number of abilities that pilots can take(6).
Just to spitball
Make a close combat/brawler tree, bring back juggernaut, and dad other abilities, like the ability to fire all your guns first(like in the tabletop game), then make a melee attack; this should be the final ability on the tree, maybe. You could do similar things with the other trees.