If RTD didn't write the character of Jack Harkness specifically with John Barrowman in mind, I'll eat my hat. I have no proof of that assertion but I believe it with every fibre of my being.
I love all the scenes you mention, Swansong and have to laugh about the stopwatch because -- all the fanwanking in the world could never convince me what the "pickup" was in that line. LOL (I'm very old and have plenty of sexual experience, TMI, but I can't for the life of me think of something really sexually appealing about a stopwatch that isn't so strained as to think, "Aren't there more obvious things to do?" LOL)
This is also speculation, but if I give the writers of S1 and 2 credit with Jack and Ianto, it's how they did their best (speculation, maybe they didn't give a rat's ass, IDK) to deal with no direction from On High (that's the speculation part atout RTD.)
How one works with two charismatic actors who can make a stopwatch-over-a-friend's-corpse sexy without any affirmative "Yes, they are in a physical relationship and like each other" (that simple, for the love of God) from the Powers What Be, is quite beyond me.
ETA: It's not so much that creative minds can't come up with something sexy to do with a stopwatch. I have faith in human creativity. I just don't think it's your standard, "Hey baby, come here often?" sort of pick up line. My response would have to be, "A stopwatch? We just met. I don't know you that well -- yet." LOL