I think them not replacing them is a way of them signalling a break from the past. A break from that old goat and his fetid, corrupt little family that tried to assassinate her character for so many years.
The ring was linked to Inaki and Steal-tina so that one, I don't think, will ever be seen again. Any other would just cause the enthusiasts to have conniptions about how many dead kittens she murdered to afford such a ring, and really, how good was it when JC and crew reigned supreme, killing elephants and cheating on his wife openly, instead of this spanx wearing commoner.
SEs have moved on. Now they're complaining how "RED" Letizia is: "She hates PP (the conservative party under Rajoy)! She's supporting PSOE (the socialists)! She's a Podemos (the extreme left party) sympathizer!" One right wing media implied that she wants to bring down the monarchy because the political coalition she supposedly favors exiled Alfonso XIII. Spanish media doesn't seem to feel a need to cite a source (a real source rather than another media report) before they report on something about Letizia. So now it's repeated by SEs all over: "The Spanx Witch is trying to exil her husband!"
As for the rings, I think she just doesn't want to wear rings anymore. She doesn't wear her gold "Leonor", "Sofia" and "Carla" bracelets either. Maybe they just got in the way when she has to shake hands (she shake a lot of hands) or do things (I don't like bracelets either because it's in the way when I work). She has other rings, e.g. the BVLGARI ring and the gold ring with a black gem (I forgot what type of gem it is). The BVlGARI ring is beautiful and has nothing to do with Inaki. They are gifts from Felipe. She doesn't wear them nowadays either.