How Do I Get Flowey Back Without Doing Genocide Or Any Other Path?

The answer you have may be a little complex. For the fact that we don't know your full progress, please keep note that I've placed story-crucial parts of the answer in that hidden message box you see. Hover you mouse over them to see the text. Anyway, Flowey is one of the few characters,

the others being Toriel, Sans, and Frisk/Chara,

who have information carried over from your last play-through in the game's files--

eg. Toriel will also not appear, Sans will ask you about your previous play-through at the beginning on the Genocide battle if you return a Genocide fight (as well as in the error screens, if I remember correctly), and Chara will remind you at the end of each play-through after that you are his/hers, and if you do many Genocide Routes back-to-back, the game will essentially lock that save, and you'll start over fresh.

The only way to have this information wiped, if I'm correct, is to find Undertale's Steam folder, assuming you're playing on PC, disable Steam Cloud temporarily, and delete the physical files from your computer, then restart Undertale quickly enough and play until the first save point in the Ruins. Then, close the game, re-enable Steam Cloud, and tell Steam to replace your Steam Cloud save file with the local system (your computer), which should then overwrite your Genocide Route file with the completely new one.

I would offer some form of video tutorial to help, but most of them either only go into how you'd edit the FUN value, or don't actually modify the save file-- and I run on a Linux computer, which means that files and folders and whatnot would be very different.

I should point out that, by my understanding, there is no way to do this or something similar to it on any console edition of Undertale, including the Switch.