Why is the Davidic promise about the bloodline rather than Solomon's reign?
The promise is about the bloodline because God said the seed would proceed from David's innermost being and He would establish his kingdom, a royal line carrying through history to the New Testament. It is not limited to Solomon's 40-year rule, and God's mercy would not depart despite iniquity.
Les explains that in II Samuel 7 God speaks to David through the prophet, saying He will set up David's seed after him, which shall proceed out of his innermost being, and He will establish his kingdom. The point is not so much Solomon's 40 years of rule and reign, but the royal bloodline that carries all the way through history until the New Testament. God says He will be a father to him and he shall be His son; if he commits iniquity, God will discipline with the rod of men and with the stripes of the children of men, as Hebrews shows an earthly father disciplines his children. Yet verse 15 says God's mercy shall not depart from Israel; no matter how far they fall short or how unbelieving they become, God chastens them time after time, but they remain under His thumb. So the covenant is about the continuing bloodline and God's enduring mercy, not merely Solomon's reign.
And when thy days be fulfilled, and thou shalt sleep with thy fathers, I will set up thy seed after thee, which shall proceed out of thy bowels, and I will establish his kingdom.