So, I think part of the problem here is the playstyle you develop overtime, playing the game. Guides, FAQs and such are great, but I'm thinking maybe just seeing how someone with more experience approaches the game would help?
It just so happens that I'm "re-leveling" Banshee (playing with Resonance! :D) and Braton at the moment - so, as an experiment, I recorded a video last night of me dealing with an Exterminate mission on Pluto (I was thinking this would be analogous to Jupiter, but challenging to me - but I forgot to take the Jupiter tileset into consideration. doh). I do appreciate that despite "raising the difficulty", that it's still, probably, much easier for me here - even if I only have 3 mods in my rank 15 (1 forma) Braton, they are Rank 9 Serration (damage), Rank 5 Split chamber (multishot) and Rank 1 Cryo Rounds (cold damage, rank 1 because I ran out of capacity! D: ), and while Banshee is squishy, I'm running a rank 10 Redirection (shields) - and of course, Banshee's second ability, Sonar (the yellow glowy spots), is picking up Braton's slack.
(Then again, I do cheat, bringing out my extremely overpowered Synoid Gammacor on the last handful of enemies - because I realized I didn't have ammo to finish them off :P)
Let me know if this helps? I could consider doing more of this if it's useful at all.
Also, I appreciate that Exterminate is a bit of a cop out - spawns are a lot easier to handle when you know that they are pretty much always in front of you :)