Why is it extremely difficult to move an already played Sim into an empty lot?

To have a Sim move use the phone "Split from Household and move". Remember to transfer enough funds to them so the Sim can afford the lot. It looks like you forgot to do this.

As jpkarlsen said, you just need to use "Split from Household and move" in the phone (under the car icon for travel) which you can do right in live mode, no need to go into CAS or use mods. Here's the steps:

Hello,

I have moved a Sim into another household using the phone. But I ended up moving the entire household. I don't know if I still have that option on my game. I can't read the tiny print. I have tried to make the fonts bigger, but I keep on getting the message in the game to make them small again. I've already have a solution for that, and I will have to wait until April to get a magnifying screen for my laptop.

However, the problem still lies that I cannot move a Sim that I split into an empty lot. Most of the options (Even on the phone) are to split and move a Sim into another household, not an empty lot. There are times in my story that I don't want to split and move them into a rental either.

I just ran a test in my own game and had no difficulty moving Vivian Lewis from the Spencer-Kim-Lewis household in Willow Creek to Hallow Slough, a vacant lot in the same world. She used her phone to Split From Household and Move. I transferred 10,000 simoleons from the household funds, which was more than enough to pay the 5,500 for her new lot and left her with 4,500 to start building and furnishing a house. I then exited to the main menu without saving. 

After reloading the same save, I used the same method to move Dennis Kim to Daisy Hovel, a vacant house that's also in Willow Creek. Again there was no difficulty at all except that I transferred 20,000 simoleons to cover the extra cost of a fully furnished starter home. The house cost 15,891 and left him with 4,109 as expected. 

I didn't bother trying to Split and Move to an apartment because PipMenace already did that with Liberty Lee. 

I don't know why you're having difficulty but I found it quite easy to Split From Household and Move to either an empty lot or an empty house. My best guess is that you just aren't giving your Sims enough money to pay for the lots you want them to move into, as already mentioned by jpkarlsen .