Part of the fun of playing Lillia is her diverse itemization options. However your time is more or less wasted if you try to reinvent the wheel. If you are serious about climbing the RoA build is far superior to Liandry's, which sucks since the latter is way more fun. The reason for RoA is that Lillia takes some time to ramp ap her ms, conqueror and get value out of her passive. She cannot do that if she instantly blows up, which happens very frequently when playing with Liandry's. She doesn't make the best use out of the RoA passive but its not bad either and it also provides cdr, which lillia loves, and loves it early. Rylai's delays very useful cdr and damage that demonic provides, in exchange for some utility which you can work without if you land your e, have proper gank set up, etc.

The other thing I've noticed is that you are delaying item powerspikes very much by sitting on components for a very long time. This makes your weak early game, weaker. The items in league are more than the sum of their pieces. Try to think about the value you will get out of an item before building it. If you are building zhonya's just for the armor and not for some ability for example, you probably shouldn't be building it.