And here's where I swoop in to half-skate around the rules instead. :P
Everyone loves controversial subjects, right? Well here's one for you, conveniently proven to at least be halfway reasonable by the PC game: non-simultaneous turns.
Getting into the nitty gritty of it is obviously for the fan rules board and not here, but I think it has some very favorable advantages, namely:
1) Player engagement and interactivity. Obviously this is one that I harp on a lot, but it's really evident when games resolve by-unit instead of by-force. The simultaneous nature of phases in BattleTech means that especially during the movement phase unless you are currently in the process of ****** up (even through no fault of your own, like trying to stand and failing) then there's not actually much happening. There's strategy, of course, but it's the strategy of a chess match, denying moves to your opponent and realizing good positions. It's a slow burn. And then everything of import happens between the shooting and physical phases, typically in one (relatively slowly resolved) swoop. And then after that it's back to the slow burn. It's very much a game of highs and lows, but the lows vastly outnumber the highs.
When each unit moves and shoots in the same activation, the strategic calculus can change at the drop of a hat, regardless of what part of the turn it is. All players have to be engaged, either because their turn can come up at any moment, or because what they were planning on doing radically shifts. It's less good (but still an improvement) in 4+ player games particularly where one team has more players than another, but the immediate engagement is still there through other moves. Why that's a bonus: I played in a campaign game less than a month ago with four players and six enemies. One turn, I was the first move in a turn we lost initiative. The next time I was required to think about the game was almost 15 minutes later when the shooting started. In between I could have happily zoned out and missed nothing of consequence. That's not possible in a game with immediate feedback on all moves.
2) Ambushes can actually happen. What happens in an attempted ambush in BattleTech? Either you invent a new rule that says the victims can't shoot or have some major penalty, or you modify existing hidden unit rules to make it so that the ambush can actually happen. Because if you don't, your "ambush" is just a non-standard deployment zone, and the simultaneous shooting phase means nothing resembling an actual ambush happens.
Tactical ambushes, clever moves you can set up for your opponent to walk into suddenly have an immediate emotional payoff. In regular BattleTech, the move completes and your opponent gets a brief moment of "oh, damn" and then it's on to the rest of the phase and then the shooting phase, potentially minutes where the emotional payoff - on both sides - of a properly executed ambush have time to fizzle. Meanwhile, when the ambush becomes apparent and then completes in the same fluid motion, the effect is immediate. The Berserker casually strolling through a level 3 building to end up one hex behind your <insert 'Mech here> and then stand there twiddling its thumbs until the physical combat phase is "That's BattleTech! Oh well, sorry <'Mech>". The Berserker casually strolling through a level 3 building to end up one hex behind your <insert 'Mech here> and then immediately cut it in half without missing a beat registers closer to "HOLY SHIT". There's no fizzle (unless it misses, in which case the payoff is still immediate, just in the form of good-natured mockery instead of mortal terror).
3) Light 'Mechs retain a use well into the late-game years. A fast 'Mech with some pulse lasers in the 3100s is just begging to get beaten by something meaner than it is because it can never, no matter how hard it tries, pull off that magic backstab and get away scot-free without some significant help. By itself a light is a nuisance, they only become dangerous when used in groups or as part of a larger Lance to make sure that it isn't focused on. It becomes wholly dependent on a lot of things, not least of which are your opponent either not winning initiative or your opponent deliberately ignoring the backstabber. Both of those things can be mitigated, but neither of them can truly be avoided with skill - it's all either luck or your opponent's mistake.
I love Light 'Mechs. In a non-simultaneous game, a Light 'Mech goes for the kill at the end of a turn, striking from behind. It doesn't score a kill (but the enemy absolutely experiences that moment of pants-shitting terror), but unlike in a simultaneous game, the 'Mech it lit up doesn't get the perfectly-informed opportunity to flip arms and make it really pay. The next turn, the backstabber sprints away, unharmed. Or potentially very harmed, if the opponent is able to respond with another well-placed move. It's now a game of play-counterplay, cat and mouse, and potentially of honest-to-god suspense. Cat and mouse games as they stand right now are more frequently than not "who won initiative this turn?" and that's the end of it.
There are definitely downsides, but this post is also really long, and it's not really a forum post if it admits weakness, right? :D