Now, as to the rest of the idea of actions, I like the concept. And, being able to sprint flat out to get 2 MP for ground but at the cost of not being able to shoot, or on the flip-side, be able to hold position and gain an extra attack for the squad, would make infantry interesting.
Daemion I like your idea of MP being the replacement for actions. So 1 MP infantry are your militia, your generic unconditioned troops who are encumbered with normal combat gear and the assumed base level armor infantry must wear to not take double damage. If its 1 MP to shoot heavy weapons, then they get to move or shoot. Rifles are simple to operate, so like the RPG you can walk and fire a rifle in a turn (with the walk penalty).
2 MP infantry could have stabilizers to represent 'well equipped' troops or quick recon troops (which are pretty cheap and one of the top 3 things to add to an autorifle or machine gun, along with upgraded scopes), or they could be elementals, or just really well trained. They can move 2 MP when they go all out, or move 1 MP and still fire their heavy weapons.
It also 'fixes' not having a run speed, which honestly infantry and motorized infantry should have. So a 2 MP infantry squad can run for 2 hexes, and if they take the normal running penalty like everything else does (including infantry in the RPG), then their rifles fired from the hip will be less accurate but still shoot-able.
It takes an action to reload the common weapons, so instead of SRM launchers dealing .57 damage cause of odd conversion stuff, they would deal 2 damage and take an MP to load. So a 1 MP infantry would be the bumbling recruits who take forever to do anything when encumbered in all their gear, while conditioned infantry with 2 MP/actions could remain stationary to shoot and reload in the same turn.
Pinning is a good word for what the RPG calls 'stunned', where you lose 1 action in the RPG, since stunned is a term for tank crews. (I think other games call a similar mechanic mechanic 'bailing' when a tank takes a pen hit and the crew cant shoot while they organize themselves?)
If we look at the actual movement rules, to see how this would line up, average joe of 4 or 5 attribute only walks 18 meters in 10 seconds, so without taking the run action they arnt moving 1 hex, but have 2 actions to use in the RPG each 5 seconds. Average joe running moves 38, or 27 if they walk for 5 seconds and run for 5 seconds, and average joe who has trained running a normal amount can move 44-46, or if they walk for 5 seconds and run for 5, they can move 31-32 hexes. So training does make the difference of being able to move 1 hex or not for average joe.
So a trained soldier (or someone really buff like an average elemental) can move 30+ meters in 10 seconds, hip firing a rifle and spending a complex action laying down suppressive fire with a heavy weapon.
A 3 MP trooper, would be someone able to make use of the sprint action. Sprinting in the RPG takes the whole turn and is fatiguing, so it takes a full turn doing nothing to remove it. Fatigue is kinda like the 'heat' thing you mentioned, and is the MASC thing I mentioned. However, for average joe you can build up 4 fatigue before the penalties kick in. You also take fatigue each attack, in addition to stun, but it only really matters if you take a LOT of fatigue (or non-lethal damage, which in battletech comes from water cannons on mech mounted fluid guns and probably a few other places).
Sprinting for 5 seconds, our untrained average joe can move 38 meters, 76 in 10 seconds if unencumbered. Thats no shooting, rifles, ect, just the same as sprinting in tactical operations for mechs and such, but with the additional limit that they cant sprint for very long. A trained average joe can sprint for 88-92 meters, so well trained high willpower troops, unencumbered or with assistive load bearing devices, can hit 3 MP in a battletech turn. Thats no actions. Such well trained, or well equipped, or good elemental troops would be able to carry more stuff. Thats either anti-mech or other specialist equipment. So when they move 2 MP they would still be able to fire a heavy weapon with a complex action, as they would sprint for 45+ depending on skill and walk for 10+ depending on equipment and buffness, hitting 60 meters in 10 seconds with enough in the tank to still lay down suppressive heavy weapon fire. Being above average, they would also be able to keep that up for 5+ turns in a row without pause, but getting hit would stun them as normal and 'pin' them/take away lots of actions.
So with the 1/2/3 MP infantry, we get more mobile troops with training, giving a reason to buy said training instead of just taking the horde of untrained infantry. The movement of 1 walk 2 run 3 sprint also lets us use the standard hit chart that mechs and the RPG infantry share for +1/+2/No shots. Motorized infantry by extension of a walk/run/sprint speed would move 3/5/6. So across the board infantry would be more mobile and more responsive on the table, and less of a 'zombie' horde. Further, a hit from a weapon would slow them down/pin them, encouraging taking actual skilled infantry instead of the no-AM skilled cannon fodder we see a lot of now. Those AM or other specialized troops would be more expensive then the fodder infantry; the fodder might not be easy to kill in bulk, but putting them under fire makes them combat ineffective without needing morale or other such rolls.
Suppressing infantry by shooting at them to pin them is a big deal in other games (and I assume real life), and this seems like a good way to implement this in BattleTech without needed any charts or rolls or such, while also encouraging the use of better trained infantry to be more effective under fire.