refactor: propagate context.Context through call chains

Thread context.Context from callers instead of creating
context.Background() at every call site. Engine stores its
lifecycle ctx for use by triggerAlert goroutines. TUI Model
carries ctx for store operations in Cmd closures. CLI commands
create a root ctx and thread it through openStore, seed
functions, and init methods.

Only two intentional context.Background() remain in non-root
positions: engine constructor default (overridden by Start)
and drainWrites (parent already cancelled at shutdown).

internal/tui/tab_sites.go

@@ -1,7 +1,6 @@
 package tui
 
 import (
-	"context"
 	"fmt"
 	"net/url"
 	"strconv"
@@ -604,10 +603,11 @@ func (m *Model) submitSiteForm() tea.Cmd {
 	}
 
 	st := m.store
+	ctx := m.ctx
 	m.state = stateDashboard
 	if m.editID > 0 {
 		m.engine.UpdateSiteConfig(cfg)
-		return writeCmd("Update site", func() error { return st.UpdateSite(context.Background(), cfg) })
+		return writeCmd("Update site", func() error { return st.UpdateSite(ctx, cfg) })
 	}
-	return writeCmd("Add site", func() error { return st.AddSite(context.Background(), cfg) })
+	return writeCmd("Add site", func() error { return st.AddSite(ctx, cfg) })
 }