In the past, in my experience, any organisation that included the British military, tends to lean on, indeed, depend on, the British soldier, (all services), and their professionalism, when something happens. Apart from the UK always willing to commit them, they are, (were?), regarded as the best trained, and fighting forces, which would manifest itself in operations.
Why have a European managed military, when NATO is already in place, alongside the UN? What would the difference be, other than letting a bunch of people who take years to make a decision, run an Army? We all know that the Russians, (again) are the main, and biggest threat. Try to imagine what would happen if things became strained, and the newly formed EU military machine lurched into action. (Whilst doing this, consider how they bodged the refugee crisis).
I guarantee that if they treated Russia like they are treating the UK, in these Brexit talks, the proverbial would hit the fan very quickly, or their defence will grind to a halt whilst they discuss things.