[Feature]: Support Logitech gaming devices — become a full G Hub alternative

### Pre-flight checklist

- [x] I searched existing issues and the [Roadmap](https://github.com/AprilNEA/OpenLogi#roadmap), and this isn't already tracked.

### Problem / motivation

OpenLogi is a great local-first replacement for **Logi Options+**, but Logitech's *gaming* peripherals are managed by a separate app: **G Hub**. Users who own both productivity and gaming gear (e.g., an MX Master for work plus a G502 X / G Pro X Superlight 2 for gaming) still have to keep G Hub installed alongside OpenLogi.

That partially defeats OpenLogi's core promise: G Hub requires a Logitech account for some features, runs a heavy Electron app plus multiple background services, and pushes cloud profile sync — exactly the things OpenLogi frees us from on the Options+ side.

Receiver-level detection for several Lightspeed receivers already works (#512), but the feature surface on top is still Options+-oriented, so gaming mice are recognized without exposing their gaming capabilities.

### Proposed solution

Extend OpenLogi's gaming-device feature set so it can fully replace G Hub, the same way it replaces Options+. Roughly in priority order:

1. **Gaming mouse basics** — remapping for all G-buttons (including G-Shift), multi-stage DPI (per-stage DPI value, e.g. 5 stages, with optional on-device color indication), and report/polling rate switching.
2. **Onboard memory profiles** — read/write on-device profiles so settings persist on other machines without any daemon (on top of #459).
3. **LIGHTSYNC RGB** — per-zone colors and effects, per-profile (on top of #351).
4. **Remaining Lightspeed receivers** — tracked in #512.
5. Later: G-series keyboards (G-keys/macros) and headsets.

It would also help to have a single "Gaming devices" roadmap/tracking item so the scattered gaming-related PRs and issues (#351, #369, #459, #730, #392, #877) have one home.

### Alternatives considered

- **G Hub itself** — what we're trying to avoid (account, telemetry, resource usage).
- **Logitech Onboard Memory Manager** — Windows-only, limited device coverage, no longer actively extended.
- **piper/ratbag (Linux)** — partial gaming-mouse support (DPI, buttons), Linux-only, no per-app profiles or richer features.

### Related area(s)

- [ ] GUI
- [ ] CLI
- [x] Button actions / remapping
- [x] DPI
- [ ] SmartShift
- [ ] Per-application profiles
- [ ] Configuration (TOML)
- [ ] Auto-update
- [x] Other (RGB / LIGHTSYNC, onboard profiles, polling rate)

### Additional context

Cross-references: #512 (Lightspeed receiver tracking), #351 (RGB effects PR), #369 (G305 + gaming features PR), #459 (onboard profiles PR), #730 / #392 / #877 (individual gaming-device reports).

For context, there is prior open-source work on talking to gaming devices over HID++ (e.g., ratbag's Logitech gaming protocol support, and the `g-hub-configurator` project for G502-style onboard profiles), which may be useful references for the protocol side.

Start by reviewing the existing gaming-device work in #351, #369, #459, #512, #730, #392, and #877, then compare the HID++ protocol references in ratbag and g-hub-configurator. This issue needs to be narrowed to a specific milestone; completion would require the selected gaming capability to work reliably for the targeted devices.

Written by the indexing model from the issue text.