First thing I noticed is your first engagement was terrible. When a pioneer charges your RE to take a fuel you need to back up and create distance. It allows you to fight from a longer range(and win) if they don't chase. If they chase you deny them the fuel. You pointlessly retreated and gave them the fuel and lost 46 manpower on the RE. This also means your rifleman lost 224 manpower due to cycling Riflemen through. If you had the 2 Riflemen and the RE semi close to each other you would forced 3 retreats and cost them a lot of ground. The engagement picking wasn't there (you effectively let them 2 v 1 you in every first engagement). If you're in a situation where you're outgunned you should soft retreat and reassess/regroup.
Your split wasn't bad, but your engagement picking is. RE and 2 Riflemen one side and the other Rifleman caps the other side and harasses is entirely an acceptable and a very strong split to open with.
You generally had the right idea pushing center(forcing a base pin attempt) and then having your engineer harass the Eastern side. However, having the engineer harassing East and the Rifleman harassing West was overextending yourself. Generally you would like to attack from as many angles as you can and neutralize the MG. The Riflemen are better than the Grenadiers so be aggressive.
Against OKW the calculus is a little different. OKW can brutalize you on the open and you want to play more to cover and MGs. This game was even more interesting since they skipped the MG which is usually good for the open for the USF player. The double pio and gren build is usually a stall move and the map control your opponent won with a cheeky stall move is scary for your prospects.
The wipe on the Grenadier was good(and gave you some breathing room). Against Ostheer(never against OKW) especially when they seem willing to pull the panzergrens(or go for any suspicious stalls) grenade tech usually makes sense. The REs and Lt can smoke which let you close the distance with the Panzergrenadiers and beat them with Riflemen.
I don't think the mortar was wise. It delays light vehicle play and you could easily flank that MG. Grab a few guys and leave the base the other way. You want to force retreats a blob flanking their MG on your cutoff will get those forced retreats.
The Captain is a very odd choice in 1v1 especially without going something like Airborne (MG drop) or Recon (M8). The Mortar and the ATG is a huge investment of manpower and throws out any light vehicle play. Even an m20 if unorthodox could be quick relief for your lack of range and sighting to counter the MG. I would have picked an m20 over both of those hands down.
Generally the build order I use is Rifle, Rifle, Rifle, Lt, M5. However between the Lt and the Stuart things can change a little. If I am low on fuel(I will often do this against OKW anyway) the MG is a great option. If I am bleeding really badly from bad engagements I will usually build an ambulance first.
Personally I would have gone double Zooks on the REs and then gone with a Sherman. The double M1 was a good call and you screened them well and that helped you hold the center. You basically turned this around with how many wipes you got. I wouldn't worry about your split it is perfectly reasonable (though contextually it can vary). I would work on picking engagements(you probably want to err on the side of more units in your attack than less). Capping (especially weird points like the munitions on Crossroads) is less of a priority than forcing retreats. If you can force an early retreat or two you can then cap after and dominate the map.
For commander I'd usually pick Armor for the M10s or Bulldozers. Both can really break up Ostheer plays. OKW I need Airborne to be on my loadoud. Third is tricky where I'd go Tactical Support, Mechanized, or Recon. You seem good with the M1s and seem to struggle with the MGs. Maybe going with Airborne would work and then you could have one pathfinder spot the MG and then have the Pathfinders crew a dropped M1 for selfspotting. If I was you I'd practice only playing Lt/Airborne for a while.
K, I get your point.
I will pay more attention to the army fighting position, to avoid the 2 v 1 situation in my early game like this.
I played Lt and Stuart before, and it feels like manpower shortage, especially, around 6-7min, right before Stuart coming out.
One more thing, I think I don't have many ideas on handling MGs as USF, and this is the reason for mortar. Somehow, I am really disappointed with the anti-infantry of Stuart.
I will keep trying on the USF, but I will say, RM is so so weak~