No, Voldemort sought only to tear down the protection that Harry had gained from Lily.
Voldemort was extremely desperate to refute the claims that he was beaten by a mere child. He would do anything to stop Harry from existing and thus would never willingly tether harry to life.
I can touch him now.
Goblet of Fire - Voldemort Talking to Death Eaters about Harry Potter
It is evident throughout the series that when Voldemort goes after power or seeks to accomplish something he often overlooks things he is extremely arrogant in that regard. In this instance he wanted to undo the protection which initially caused his downfall. He found a solution and did not bother to look further into it, his arrogance leading him to believe that what he thought was correct. He is rarely seen (if ever seen) to second guess his choices, where a wiser wizard may take a step back and research he starts to plan how to accomplish his newly found solution through any means necessary.
if Voldemort were to try to kill him for a second time?
Well this is largely circumstantial, after Harry is killed the first time he sacrifices himself causing the same protection that Lily granted him. Voldemort does try to kill Harry during the second phase of the Battle of Hogwarts but his curses do not stick. Example: The body bind curse placed on Neville.
Would this tethered connection still exist?
The sacrificial magic used makes it impossible to know, as we'd have to draw a lot of assumptions about things that have no other evidence in canon. The fact is that from the moment Voldemort kills Harry in the forest his fate is sealed.
If he'd struck Harry in the Graveyard during GoF the same events would follow as from the forest scene in DH. However, there would be no sacrificial magic involved so we have no idea what would happen.
From the sounds of the other Answer here and the referenced Answer in the other question, along with your quote. It sounds like Voldemort turned himself into a makeshift Horcrux for Harry. Whilst Voldemort still existed Harry was tethered to the living world. In which case Voldemort would never be able to "kill" Harry whilst he was himself alive.
Obviously it is not a complete Horcrux it seems more complex and less dark of a creation, using merely the blood rather than the indescribable dark magic required for a Horcrux. If it were a complete Horcrux we'd have the whole Voldemort kills the Horcrux in Harry but in reverse when Harry killed Voldemort.
TL:DR read the first line.