At present, it's a requirement of the Single Market that all car insurance policies have 'whole-EU' coverage, to unified standards of what that means in terms of minimum cover - so it makes no difference what it says or doesn't say on your policy docs, you are assumed to be covered. Sorry if I'm stating the obvious.
After 29 March (if no-deal) the UK becomes a 'third country' and I could well imagine police requiring to see a green card... just as the British police would if you drove, say, an Algerian car in the UK regardless of what was written on the Algerian policy docs in whatever language.
I know that sounds bureaucratic but, personally, I wouldn't think of going onto the continent after 29 March (if no transition period in effect) without an international green card, as I wouldn't want the hassle of spending a couple of hours in a French or Italian police station proving I was actually insured. Just my view.
[EDIT] Actually there will be a lot of media spotlight on the charges UK insurers decide to make to issue green cards so I could imagine they might decide to waive the charge as a 'free benefit' of policies. [EDIT EDIT] I only said 'might'!