This would happen anyway with ppshs. Cons spam is not an issue of weapon upgrades its about upkeep/mp drain. Only difference is that ppsh should be for assault and dp28 for defense, so hit the dirt fits on static defensive weapon profiles.
Blobing PPsH conscripts far less viable than blobbing DP-28 conscripts. The reason is that PPsH Conscripts have very low DPS at long range. If they manage to activate Hit the Dirt so before they get suppressed by the MG, their DPS is so low that the MG can walk away. If they were all armed with DP-28s, the MG would have been shot to bits.
Imagine LMG42 Grenadier blobs (already problematic), except that they're much harder to wipe with indirect fire, they have sprint, and they're immune to suppression and take less damage.
I understand your point, that you would rather have something that is thematic and synergizes extremely well with the ability. However, I would argue that such a combination would make Conscripts overpowered.
If you're so inclined, feel free to make a mod. Give conscripts a DP-28 upgrade, pick a commander with Hit the Dirt, then test it to see how they perform. I'm pretty sure that you will make a unit that is incredibly frustrating to play against, and fairly easy to a-move in a blob.