fixes #35898

src/vs/workbench/parts/debug/browser/debugStatus.ts

@@ -54,8 +54,9 @@ export class DebugStatus extends Themable implements IStatusbarItem {
 				this.quickOpenService.show('debug ').done(undefined, errors.onUnexpectedError);
 			}));
 			statusBarItem.title = nls.localize('debug', "Debug");
-			this.icon = dom.append(statusBarItem, $('.icon'));
-			this.label = dom.append(statusBarItem, $('span.label'));
+			const a = dom.append(statusBarItem, $('a'));
+			this.icon = dom.append(a, $('.icon'));
+			this.label = dom.append(a, $('span.label'));
 			this.setLabel();
 			this.updateStyles();
 		}

src/vs/workbench/parts/debug/browser/media/debug.contribution.css

@@ -103,8 +103,8 @@
 }
 
 /* Debug status */
-.monaco-workbench .part.statusbar .debug-statusbar-item {
-	cursor: pointer;
+/* A very precise css rule to overwrite the display set in statusbar.css */
+.monaco-workbench > .part.statusbar > .statusbar-item > .debug-statusbar-item > a:not([disabled]):not(.disabled) {
 	display: flex;
 }