I rarely say this but this episode was rather pointless. The reason being.... If they had agreed to work with Root five episodes back then Root could easily solo this mission and it would take her about 15 min total, no casualties, no excessive force. and wham bam thank you ma'am done!!!!
The reason it would be simple is Root has a different relationship with The Machine than Finch does. Whereas Finch has only one purpose for The Machine-- saving lives. Root sees The Machine as alive and an accomplice. Root has asked the Machine in the past for additional information and to aid her in her time of need. The machine does a system check to see if any favors involved harming/taking human life and if the favors don't harm people---- it helps Root out.
So back to what I am saying, Finch's moral code prevents him from trusting Root or allowing her to join the team. Plus Root is one of my favorite characters.