Simply, Jesus is our sole Mediator.
Jesus is not our sole Intercessor. The rest of the Timothy proof-text, read in context, makes that clear:
**I urge, then, first of all, that requests, prayers, intercession and thanksgiving be made for everyone — ** for kings and all those in authority, that we may live peaceful and quiet lives in all godliness and holiness. This is good, and pleases God our Savior, who wants all men to be saved and to come to a knowledge of the truth. For there is one God and one mediator between God and men, the man Christ Jesus, who gave himself as a ransom for all men—the testimony given in its proper time (1 Timothy 2:1-6)
Jesus is our sole Mediator in that He was able to atone for our sins, as He is both God and Man - no one else could bear the weight of our sins, and this was His role as Mediator, as St. Paul acknowledges in the same sentence. But the rest of the text the OP quoted shows that Paul called us all to act as intercessors, even as we ask the saints to be our intercessors (and even as Abraham interceded with God for Sodom, saving Lot in the process) - so for Protestants to try to use this as an indictment against asking for the prayers of others, living or dead, has always been fallacious.
A Mediator is one who resolves disputes between two parties (in this case, the two parties to the Covenant between God and Man). An Intercessor, in the Christian sense, offers prayers to God on behalf of others. Any member of the Church, living or dead, can do the latter but not the former.