When I put in a new plant, like an Amazon sword, that have huge roots I trim the roots up some until they are about 3" in length. Then I plant it. It helps the roots grow and establish better. My huge amazon that has been in the same spot for awhile has huge roots that come right out of the sand and into the water column. One root is even taller than the plant! No, roots sticking out into the water is fine.
Did you rinse the baby tears under the sink before you planted it? That should have gotten the packing stuff off the roots. I don't think that is the reason why it is not growing. Do you use any kind of plant fertilizer?
I use plant fertilizer, CO2 and high lighting and for some odd reason I cannot grow stem plants like baby tears. They just slowly die off on me. There are only a couple kinds that will grow. Must be my water supply or something. I have yet to figure out what.