Unable to add keys with ssh-agent plugin in zsh

I just installed arch, and I am trying to configure my shell, zsh. I am currently using zinit as my plugin manager with powerlevel10k. I was trying to automate the activation of the ssh-agent and the addition of keys to it. This is my current .zshrc file:

neofetch

# Enable Powerlevel10k instant prompt. Should stay close to the top of ~/.config/zsh/.zshrc.
# Initialization code that may require console input (password prompts, [y/n]
# confirmations, etc.) must go above this block; everything else may go below.
if [[ -r "${XDG_CACHE_HOME:-$HOME/.cache}/p10k-instant-prompt-${(%):-%n}.zsh" ]]; then
  source "${XDG_CACHE_HOME:-$HOME/.cache}/p10k-instant-prompt-${(%):-%n}.zsh"
fi

# Set the directory to store zinit and plugins
# ---  Defined in the .zshenv file

# Download Zinit if it's not there yet
if [ ! -d "$ZINIT_HOME" ]; then
    mkdir -p -- $ZINIT_HOME:h
    git clone https://github.com/zdharma-continuum/zinit  "$ZINIT_HOME"
fi

# Source/Load zinit
source "${ZINIT_HOME}/zinit.zsh"

# Add in Powerlevel10k
zinit ice depth=1; zinit light romkatv/powerlevel10k

# Add in zsh plugins
zinit light zsh-users/zsh-syntax-highlighting
zinit light zsh-users/zsh-completions
zinit light zsh-users/zsh-autosuggestions
zinit light Aloxaf/fzf-tab

# Add in snippet
zinit snippet OMZP::git
zinit snippet OMZP::sudo
zinit snippet OMZP::archlinux
zinit snippet OMZP::command-not-found
zinit snippet OMZP::ssh-agent

# Load completions
autoload -U compinit && compinit

# Replay all cached completions ( recommended by documentation )
zinit cdreplay -q

# To customize prompt, run p10k configure or edit ~/.config/zsh/.p10k.zsh.
[[ ! -f ~/.config/zsh/.p10k.zsh ]] || source ~/.config/zsh/.p10k.zsh

# Keybind - emacs style, enables ^f ^b ^a ^e ...
bindkey -e
bindkey '^p' history-search-backward
bindkey '^n' history-search-forward

# History
# -- some changes already in .zshenv file
setopt appendhistory # append to history file instead of overwritting
setopt sharehistory  # share command history between sessions
setopt hist_ignore_space # ignore commands starting with space 
setopt hist_ignore_dups
setopt hist_ignore_all_dups
setopt hist_save_no_dups
setopt hist_find_no_dups # Don't show duplicates in history search

# Completion styling
zstyle ':completion:*' matcher-list 'm:{a-zj}={A-Za-z}'
# Set the default colors to LS_COLORS
eval "$(dircolors -b)"
zstyle ':completion:*' list-colors $LS_COLORS
zstyle ':completion:*' menu no
zstyle ':fzf-tab:complete:cd:*' fzf-preview 'ls --color $realpath' 
zstyle ':fzf-tab:complete:__zoxide_z:*' fzf-preview 'ls --color $realpath' 
  # ssh-agent configurations
zstyle :omz:plugins:ssh-agent quiet yes
zstyle :omz:plugins:ssh-agent lazy yes
zstyle :omz:plugins:ssh-agent agent-forwarding yes
zstyle :omz:plugins:ssh-agent identities ~/.ssh/github_rsa

# Aliases
alias ls='ls --color'
alias c='clear'
alias b='cd ..'

# Shell integrations
eval "$(fzf --zsh)" # Fuzzy finding
eval "$(zoxide init --cmd cd zsh)"

As you can see, for ssh-keys I am using: zinit snippet OMZP::ssh-agent and the following settings:

zstyle :omz:plugins:ssh-agent quiet yes
zstyle :omz:plugins:ssh-agent lazy yes
zstyle :omz:plugins:ssh-agent agent-forwarding yes
zstyle :omz:plugins:ssh-agent identities ~/.ssh/github_rsa

~/.ssh/github_rsa is the location of a valid key and has no password associated with it. My ~/.ssh/config file is super simple. It only contains the line AddKeysToAgent yes. ---- Initially the problem I was facing was that no key was added to the agent. Now, after some testing I found that the key only gets added to the agent if it has the default name id_ed25519, otherwise nothing happens ( at least with the values I tested for ) . I couldn't test the loading of more than one key since they need to have different names, which means, different from the default which by my tests, doesn't work. As suggested, I tried removing the setting zstyle :omz:plugins:ssh-agent lazy yes even if not recommended when using powerlevel10k. Weirdly enough, if the key has the default name it gets loaded anyways