So I'll weigh in as someone who has not played any other version of Traveller outside of a small amount of hand-picked rules from a couple other systems that I liked for MgT2E.
I really like the combat system as it sits. There's a lot in this new material that I don't care for as it is written right now, but I see some things that might be good with some tweaking.
Initiative: Wholly opposed. Full stop. By placing initiative solely on the shoulders of one player, the entire scene can be disastrous if they get a bad role, and not just for them but for everyone. This also comes into an issue of why would anyone else bother to build up their tactics when Commando Bob already has a +3? The last issue is the "pick an action and commit." Which reminds me of the old Final Fantasy game on the original Nintendo. If you had two or more characters target the same enemy and it died after the first hit, the other moves were wasted, which quite often could lead to a total party kill. There's a reason this went away 40 years ago, let's not bring it back now for the sake of "realism" in a game that is anything but real. Sorry, but I'm pretty hard core set against this and will not use it for any reason. Punishing another player who "chose poorly" after the first player managed to get a lucky hit is just bad design.
Melee: The 2+ for someone not dodging or parrying is stupidly low, and if you combine it with issue #1, suddenly you have a scenario where players (or enemies) can run up on their initial attack and get a free hit. It's somehow worse than "flat-footed" in other systems as there's nothing here to establish them being flat-footed beyond, "Well, you don't get to go first so sucks to be you." Their only option is to take a DM penalty for their turn, simply because the player that rolled initiative botched it. Once again we're back to punishing players. It needs to stay at the normal 8+.
Pistols in melee has already been addressed by plenty of others, I can only echo their sentiments, especially as a person wo has extensive knowledge, practice, and real-world applications of firearm tactics.
Charge: I like the idea, the implementation needs work. The triple-move combined with the melee attack is too much. I would limit it to a double-move. Also, I don't think it should get a damage increase. First, if someone is running at you, they are broadcasting their action and while they might be likely to make contact with you, unless you are just blissfully unaware of this attack, you have enough reaction time to brace for any impact. In fact, a skilled grappler could and most likely would simply use their own momentum and toss or deflect them. That's also why I am on the fence about the +2 DM. It would make it more difficult to dodge when it should actually be easier. I want to play with this some though before completely dismissing it. On of the issues currently with melee is closing the gap.
Disarm needs to stay a grapple move. Sure it looks fancy in the movies, but a hand is a remarkably strong machine and a rather small target to hit with enough accuracy to disarm someone. As many want to point out, combat in Traveller isn't supposed to be like the flashy movies (but flashy is inherently more fun.) The other issue combines with the "flat-footed" reference I made above. Suddenly disarming someone is an almost guarantee unless they specifically dodge or parry. If they have no dodge or parry abilities, that's a recipe for disaster. I'm curious if this is also able to be combined with Charge?
I like the dual wielding. I'll have to play with it to see if it's still too easy to hit. Also, what about fists? Are those considered two weapons?
Multiple actions: I could see this becoming standard after some balancing passes.
Multiple Enemies: I think it highlights the issue with the new initiative change and melee attack difficulty especially if you combine it with multiple actions. Two enemies could get up on me (even if I have melee skills) and I am locked down with their opening salvo. First guy does a double attack. I can try to dodge both, but now I'm at a -2 DM on my turn, even if he misses. I don't get to make that decision after his attack roll. The second guy does the same, so now I'm at a -4 DM. Even if by this point I haven't been hit (unlikely.) To avoid that onslaught would come if I did not react, I had to completely cripple my ability to fight back in the next round. I can only attack 1, while starting at a -4 DM, and if I try to flee, they each get a free attack, starting the cycle all over again. Add in the new charge and even if I take a triple move, I can't get away. Just 2 enemies could keep me completely locked down unless initiative is lucky enough to change. I'm effectively useless to the group. That's not fun, it's frustrating.
On to the AP changes. I'm going to say the thing no-one else is saying right out of the gate: It's pretty shitty of you. The CSC just came out. It's not even a year old yet? Roughly? And it's not an inexpensive book. Your proposing a change that would radically alter a large amount of what's in it, and you'll give us an errata pdf? Do you have any idea how much of a pain in the ass that would be when your at the table and someone wants to buy a new piece of armor or weapon? Something so drastic is a next edition change, not a change that invalidates hundreds of entries in books that already cost a lot of money. You don't do that! Screw the merits if there are any, you're screwing your customers that paid good money for stable material. And don't even try to argue "just don't use them." If that's the case, then why implement them at all? And once alternate rules are out there, some players will get rather cranky if you try to tell them no. If you can't tell I'm a little sore about the idea of my shiny new book being a decoration as I'd have to print out many pages of errata to hand out.
With that out of the way, my issue is I feel like this will invalidate other ammo types and certain armors. For clarification, are the AP amounts in the document using AP ammo, or default values with AP ammo providing even more of a bonus? It reads as if they have built-in AP. The current system does a better job of representing how actual soft armor currently works. My level 3A ballistic vest will stop the slug of a 12 gauge 1 oz slug shot, but it's going to hurt like hell, bruise and probably break a rib. It might even do enough traumatic damage to kill me, even though the armor technically stopped the slug. Ballistic vests aren't the best choice against a stabby weapon. A knife will go through it. It will slow down the impact considerable and keep it from going as deep. It will also keep the blade from twisting and doing further damage. The current rules feel appropriate to that. According to the way the proposed changes read, that knife (or equivalent weapon and armor based on tech levels) would pierce through and do the full damage. As far as "simplifying" the math, this isn't really one of the areas where that's a difficult math check. The proposed dodge mechanics are more complicated than the current AP system.
That's my rant. I'll give a better opinion once I have time to try the systems.
EDIT: I just saw the follow-up post from Matt where he says these changes would be years down the line and incorporated into the next edition ruleset if implemented. I apologize it I was a bit critical. It's good to know you recognize the issue.