I would like to add spells to another classes spell list, and I was hoping to do so using a feat that could be put on the character. I however realised that adding spells to a feat would make the character learn the feat instead of just having it as an option when learning spells. Is there something I'm missing or is it just not currently possible?

I was hoping there would be a faster way to do it, but I guess it's down to that. Thank you for the response.

You can add spells to lists via a homebrew background.

I found what you were referencing, however I have not been able make it work.

When you create a homebrew spell, it lets you use an existing spell as a base that you can make changes to. 

Say you wanted entangle on a arch fey warlock.  You just have to go to homebrew, create spell, choose entangle as the base, and have the only change you make be that it's on the subclass spell list.  

Easy day!

You could also homebrew a copy of their subclass and add as many spells as you like to their spell list.

Quote from DxJxC >>

You can add spells to lists via a homebrew background.

 How though? When I do custom background I don’t get spell list options 

Quote from Jackc19 >>

Quote from DxJxC >>

You can add spells to lists via a homebrew background.

 How though? When I do custom background I don’t get spell list options 

 Not “custom background,” you have to create an entire homebrewed background.

https://www.dndbeyond.com/homebrew/creations/create-background

You can now also do it with homebrewed subclasses too.

https://www.dndbeyond.com/homebrew/creations/create-subclass