### System details
Intel Core i5-10400, Intel UHD 630, Omarchy 4.0.0-1, Hyprland 0.56.2, Chromium 151 / Zen Browser (Wayland native)

### What's wrong?
Selecting an emoji from the emoji overlay (`SUPER + CTRL + E` → `omarchy.emojis` → Enter) inserts correctly into terminals (Ghostty/etc.), but often does nothing in browser text fields (Chromium, Zen/Firefox).

**Steps to reproduce**
1. Focus a text field in Chromium or Zen
2. Press `SUPER + CTRL + E`
3. Select an emoji and press Enter
4. Observe: emoji does not appear in the browser field
5. Repeat the same flow focused in a terminal → emoji inserts as expected

**Expected**
Emoji is pasted into whatever app had focus before the overlay opened.

**Actual**
Insert works in terminals; browsers stay unchanged (emoji may still be briefly on the clipboard).

**Root cause**
`omarchy-menu-emoji-insert` always pastes with `wtype` + `Shift+Insert`:

```bash
wtype -M shift -k Insert -m shift
```

That matches terminal paste bindings, but browsers expect `Ctrl+V`. Omarchy already teaches this distinction for `SUPER + V` in `default/hypr/bindings/clipboard.lua` (terminals → Shift+Insert, otherwise → Ctrl+V via `send_key_state`). The emoji/clipboard paste helpers never did.

The same Shift+Insert-only path is used by `omarchy-clipboard-paste-text` / `omarchy-clipboard-paste-file`, so the clipboard manager has the same browser gap.

**Diagnostics**
- `omarchy-debug --no-sudo --print` written to `/tmp/omarchy-debug.log` on this machine
- Active browser windows are Wayland-native (`xwayland=false`) with `chromium-based-browser*` / `firefox-based-browser*` tags; terminals carry `terminal*`

**Fix**
PR incoming: add `omarchy-cmd-send-paste` that mirrors `SUPER + V` (terminal tag → Shift+Insert, else Ctrl+V; prefer Hyprland `send_key_state`, fall back to `wtype`) and call it from the emoji/clipboard paste helpers.