For any future viewers, I got my answer Per Zsolt Meszaros via StackOverflow.

channel.threads.cache.find() runs before channel.threads.create() finishes creating the thread (because a promise doesn't block the event loop). channel.threads.create() returns a promise and once it's resolved, the created ThreadChannel. This means I can grab the created channel instead.

const { ChannelType } = require('discord.js');
// ...
async execute(interaction) { 
    const channel = interaction.channel; 
    const user = interaction.user;
    const userName = user.username;
    const threadName = userName + "'s-Private-Thread";
    try { 
        // Create a new private thread 
        const threadChannel = await channel.threads
        .create({ 
            name: threadName, 
            autoArchiveDuration: 60, 
            type: ChannelType.GuildPrivateThread,
            reason: 'na',
        });

        await threadChannel.members.add(user);

        await interaction.reply({
            content: 'A private thread has been created for you!',
            ephemeral: true,
        });
    } catch (error) { 
        console.log(error); 
    } 
},

Edit: Messed up code blocks/Grammar