RAISE THE MOD CAP

Get better mods, I bet you have a ton of awful ones with low or no speed or mismatched sets, like defence mods with crit damage or offense or crit damage with defence primaries. Keep improving your mods, slice, splice and 6-dot them, don't just grow the hoard. Quality over quantity. Most characters can be permanently modded and left to do the job, replace bad mods with good mods and then even better mods as you acquire them. 
You have a hoarding problem. 500 unassigned mods is just dumb. Put them on your characters. That's what they're there for. 

While 500 should be plenty for most players, what we really need is mod loadouts.  We have 300 characters, but only 200 mod loadouts, which doesn't even take into account different events we'd use different mod sets for, like TB, TW, assault battles, etc.  

If you have any mods with less than 10 secondary speed by "blue" delete them.  Waste of time and space to keep them.  If you don't have characters modded, add mods to them except the most worthless at the bottom you aren't doing anything with.  But if they'll be something some day, put the mods on them.

500 unassigned mods .............   
6 Mod slots per character = 80+ full character sets.
8 different mod "sets" = 10+ Mods for every single set * every single slot.

Here's an idea.
Go into mods, select speed secondary, sort by speed, flip order from Low to High..........  START DELETING

Sorry, that probably came off as harsh.
Honestly, At one point I think we were all capped.  (God knows I was)
But seriously, 500 in an insane # of space once you actually CLEAN OUT YOUR MODS.

Yes, it is time consuming.   Your going to eat up an entire Saturday afternoon every few months  (I'm NOT exaggerating)
But if you clean out the bank to 300 or so Unequipped & then do small weekly cleanings as you level/slice it will take several months to get back into the 500 range again.

Raid swaps can be annoying,  I suppose,  if you do all 5 runs w/o ever fixing the teams that already ran.
I learned that by the time I went into Run 3-4, that I should go re-equip my Run #1 team mods.
Remod for TW?   It locks the mods in place like GAC doesn't it?
Who/What are you remodding for TW that isn't robbing you of mods from another team?
TB I could see I suppose if I was actually concerned about that.
RotE missions are worth such small points compared to ops/deploy that I don't even bother to remod for them.

Raise the mod loadout cap while they're at it. 200 loadouts vs 300 characters... now that's archaic and outdated.