OK, risking that this might be considered a pet peeve of mine: contrary to widespread belief, MP43/MP44/StG were produced in sufficient numbers to be used in concentrated quantities by quite a number of units. There's enough primary source material that shows this and enough (mostly German language) secondary sources describing it. Also, deliveries had usually not much to do with the unit being considered as 'elite' (be it at the time or rather in post-war times).
In fact, this weapon, for the first time combining lightweight automatic firepower and rifle accuracy and range, brought along a whole new tactical concept: the MP-Zug, later renamed, along with the change of the weapon's designation, to Sturm-Zug ('assault platoon'). The new concept was to replace the previous basic tactical concept of the German infantry, which had evolved around the squad's le.MG with support of the riflemens' Kar98k. The final goal was to fully replace both the MP, the Kar and originally also the le.MG of all rifle platoons with the MP43/MP44/StG, something which was envisaged to be accomplished by the earliest somewhere in late 1945.
Deliveries of the MP43/MP44/StG44 were twofold:
1. the official ones, assigned in great quantities to dedicated units by the Organisational section of OKH. After initial troop trials and once production had reached sufficient quantities, delivery in earnest was started in the beginning of 1944, whereby priority was given to infantry divisions on the Eastern Front: their Grenadier companies, in desperate need of any boost in strength, were to receive enough weapons so they could each completely equip one of its three platoons. The aim was to gradually equip every infantry division in this way, one after another. Newly established divisions in the Summer of '44 (29. Welle) also received enough weapons to equip one platoon per rifle company. The Volksgrenadier-Divisions of the 32. Welle however were to even receive the double quantity, allowing for two out of three platoons to be equipped as such. Actual deliveries in big quantities did happen, as strength returns of Autumn 44 show.
2. deliveries in smaller quantities, meant as individual replacements for other regular SMGs like the MP40. The infantry branch wanted to prevent such deliveries since they caused both the loss of the advantage of employing the weapon in concentrated quantities and problems in ammunition supply. However, the General Quartermaster was confused by the weapon's designation as MP43/MP44 (Maschinenpistole - SMG) which did not make clear its different tactical capacities. This name had originally been used to mislead Hitler into allowing the weapon's production, in spite of his disapproval because of the logistic difficulties in producing a new model of infantry ammunition.
Now, the official MP/StG-Züge were never to have been equipped with 9mm SMGs. The concept is completely linked to the development of the new weapon. The misunderstanding (which found its way into the US Army's Handbook on the German Army) is very likely also a simple consequence of the mere fact that the StG was first called MP, and the KStN (TOEs) of 1.9.44 establishing the MP-Züge in the Volksgrenadier-divisions did so without specifying the weapon type (i.e. which model of MP, as this detail was not added to the KStN-templates): read like that it could be just any MP. However, all other contemporary accompanying documents (manuals, formation orders...) leave no doubt: the concept of the MP-Zug was there to be used with the MP44 from the start (Later the distinction was added within the KStN, by using square brackets to show StG).
However, wanting to further spread the positive effects of the mere concentration of firepower in the MP-Zug, OKH in October 44 ordered units which had not yet received their big deliveries of StG to already start making similar makeshift assault (reserve) platoons by concentrating as much as possible any MP, small number of StG and semi-automatic rifles (K41 and K43) on hand.
It is likely that this make-shift order was gradually also carried out by Panzergrenadier units, which until then had not received large deliveries of MP44/StG44. This was the case because the PzGrenadiere did not depend from the Infantry branch, had their own different tactical doctrine (around two leMG per squad) and with the added firepower and shockpower of the accompanying armour on hand were considered to be less of a priority for the rearmement. The only exception to this were the Pz.Brigades 105 to 108, formed in August 44, which had a specific set of KStN specifying the use of the MP-Zug. Later on, the Sturmzug would finally be established into the motorised Panzergrenadier units, with one such platoon per company. Although the KStN concerned were already dated 1.11.44, the general order for reorganising along these lines actually only came in early 45.
Finally, what did an official MP-Zug / Sturm-Zug in a Grenadier-Kompanie look like? The first version of early '44 consisted of three similarly equipped squads, with every man equipped with the MP43, except for those riflemen equipped with the rifle grenade launcher and the scoped weapon in each squad, who both kept these weapons. No lMG's remained with the platoon. After some adaptions due to combat experience, the final version of late 44 reinserted the three le.MG as they were still deemed necessary: one remained in company reserve, and two were being carried into combat by the platoon's third squad, the so-called Feuergruppe. The two other Gruppen were entirely equipped with the StG44. The rifle grenade launchers were concentrated at Platoon HQ level, and the sniper rifles at Coy HQ level. Except for concentrating firepower, this organisation also facilitated command and control in the meagerly trained late-war units. It was this version that was retained for the model organisation 'Infanterie-Division 45' to which all existing divisions were to reorganize in early 45. If the weapons and ammo were available that is...