Instead of a resupply crate after every objective allow players to refill ammo/consumables (maybe not grenades/rockets etc) from backpacks. Youre being deployed via a truck so a large pack like that makes no sense unless it could be used for something.
Using packs could also mean potentially ditching it at an objective allowing for faster movement and a resupply point when running low on ammo, then picking it back up to move to the next objective. Or if you are so inclined allow buddy mags letting other team members resupply from the pack on your back, if they use the same calibre/weapon. Of course using packs would come with a weight cost associated with it meaning you are almost guaranteed to be over the limit until its empty or off.
Ditching supply crates could also mean the introduction of medic class/medical supplies like IFAKs and larger medical consumables. Maybe even a medic class capable of carrying multiple kits etc. This could also mean an addition of bleed out mechanics where you could die from trauma unless you apply a kit to stop said bleed.
So many possibilities could come from removing the crates and letting people carry the ammo they need for the whole round. Potentially swap classes from a destroy able ammo dump objective for example.