diff --git a/pkg/rpc/peer.go b/pkg/rpc/peer.go
index 3f85fcf1b0b6..5533bead8d6e 100644
--- a/pkg/rpc/peer.go
+++ b/pkg/rpc/peer.go
@@ -240,19 +240,15 @@ func (p *peer) launch(ctx context.Context, report func(error), done func()) {
 	if err := p.opts.Stopper.RunAsyncTask(ctx, taskName, func(ctx context.Context) {
 		p.run(ctx, report, done)
 	}); err != nil {
-		// Stopper draining. Since we're trying to launch a probe, we know the
-		// breaker is tripped. We overwrite the error since we want errQuiescing
-		// (which has a gRPC status), not kvpb.NodeUnavailableError.
-		err = errQuiescing
-		report(err)
-		// We also need to resolve connFuture because a caller may be waiting on
-		// (*Connection).ConnectNoBreaker, and they need to be signaled as well
-		// but aren't listening to the stopper.
-		p.mu.c.connFuture.Resolve(nil, errQuiescing)
+		p.onQuiesce(report)
 		done()
 	}
 }
 
+// run synchronously runs the probe.
+//
+// INVARIANT: p.mu.c is a "fresh" connection (i.e. unresolved connFuture)
+// whenever `run` is invoked.
 func (p *peer) run(ctx context.Context, report func(error), done func()) {
 	var t timeutil.Timer
 	defer t.Stop()
@@ -266,14 +262,9 @@ func (p *peer) run(ctx context.Context, report func(error), done func()) {
 			return
 		}
 
-		// NB: we don't need to close initialHeartbeatDone in these error cases.
-		// Connect() is cancellation-sensitive as well.
 		select {
 		case <-ctx.Done():
-			// Stopper quiescing, node shutting down. Mirroring what breakerProbe.launch
-			// does when it can't launch an async task: leave the broken connection around,
-			// no need to close initialHeartbeatDone, just report errQuiescing and quit.
-			report(errQuiescing)
+			p.onQuiesce(report)
 			return
 		case <-t.C:
 			t.Read = true
@@ -700,6 +691,17 @@ func (p *peer) onHeartbeatFailed(
 	p.ConnectionFailures.Inc(1)
 }
 
+// onQuiesce is called when the probe exits or refuses to start due to
+// quiescing.
+func (p *peer) onQuiesce(report func(error)) {
+	// Stopper quiescing, node shutting down.
+	report(errQuiescing)
+	// NB: it's important that connFuture is resolved, or a caller sitting on
+	// `c.ConnectNoBreaker` would never be unblocked; after all, the probe won't
+	// start again in the future.
+	p.snap().c.connFuture.Resolve(nil, errQuiescing)
+}
+
 func (p PeerSnap) deletable(now time.Time) bool {
 	if p.deleteAfter == 0 {
 		return false