RE: 16 month old not interested in walking
you can also use something like a hula hoop, yard stick, etc. If he's not yet willing to walk with hands held a little lower than you (OP) have been doing, he might not be ready for this, but you can try it.
A lot of kids walk better with an object in their hands in general.
I've had kids that would NOT let go of having support even once they were probably very capable. I had a colleague show me a trick where she actually used a necklace (like Mardi Gras type beads) and had the child hold onto the necklace while she held the other end. Weird, but it sometimes works to wean them off the need to have a hand held.
To the OP, those are things to try once your son is receptive to having his hands lowered a bit or walking with one hand held.
edited to fix typo
Mama to DS-2004
DD-2006
and a new addition-ds born march 2010