Seeking Advice on This Houseplant

My Grandma Weaver's house was full of houseplants, many of them given to her from owners who gave up trying to keep the plants happy. The entire wall in her daylight basement housed babies and big ones of all kinds, nurtured in plastic margarine tubs, yogurt cups, or old plastic nursery pots. When she passed on to glory 20 years ago, I was thrilled to get two of her houseplants.

My sister and I shared the jade plant and have propagated many baby jade plants successfully over the years (Locals! You can always hit me up for a baby jade plant). 

However, I am baffled by the spiderwort plant I have. It has nice looking pink and green tips, but all these papery dead leaves on the rest of the stems. I have tried re-potting, different light, re-propagating and starting over.. . . but it always reverts to this disheveled look with pretty tips.

It still looks strange even with the dead leaves removed.  I'm loathe to ditch the spiderwort because it's a direct connection to a grandma I cherish. . . but I'm not pleased with its looks. Any advice for getting rid of those dead-looking stems/leaves? For getting it to look less scraggly?

I think partly what is going on is that perfectly-shaped houseplants in trendy pots are terribly popular right now. This is testing my loyalty to my houseplants that are less cute. Also, spring is springing and I feel the urge to clean up, freshen up, and refurbish all the things.