There are no experts on this Troy and people are still feeling their way with intelligent questiones like yours guiding their path.
My own limited knowledge suggests that once yo uare burning fossil fuel in air you need to carefully consider your MVHR system and it limitations.
The Mechanical Ventilation and Heat Recovery system [MVHR] should be able to both supply air to the gas hob and extract the water vapour this will generate.
When I say "should" I don't mean thay all can, but that if you're using an MVHR system AND a gas hob, it'll need to deal with the prodiucts of combusion and cooking.
Cooking products will be generated in addition to the water vapour generated by the gas burning in air, and any steamy kitchen will show these are considerable.
Otherwise the gaseous products of cooking may need to be vented to air.
The supplier of the sustem should be able to confirm/deny what I say.
In terms of open gas fires for living spaces, I think simething like that would overload the system and leave a hole through the house that would ruin the internal efficiency.
Restrictions on open fires suggest they are usually kept behind sealed glass units and a gas fire may need a balanced flue air feed/chimney from external air.
At the rate we're going we may end up living in super insulated houses and cooking on bar-be-ques on the patio.
ONQ.
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All advice on AAM is remote from the situation and cannot be relied upon as a defence or support - in and of itself - should legal action be taken.
Competent legal and building professionals should be asked to advise in Real Life with rights to inspect and issue reports on the matters at hand.