The excerpts I quoted are Grey's, from Grey's memories. I see no reason to suppose he was lying.
I don't have any issue with your source material, it is more your interpretation of it. You reference material has Grey being very dissatisfied with the prospect of a neutral Britain, but seeing nothing to gain by resigning, which coincidentally reconciles with my assessment of the facts. However, you then conclude Grey would metaphorically burn down the Liberal house. Sorry, but I cannot see any nexus between your reference material and your conclusion.
Grey was clear at the time and later he would go - the notion he'd have remained in a non-interventionist British cabinet is fictional. What Grey never explicitly stated was that he would bring down the Liberals after resigning, but in his memoires, IMO, he hints at it.
In the first instance, the absence of the Germans in Belgium means British neutrality is very much the mainstream position. What you are suggesting is you cannot envisage Grey existing in any British Cabinet not hell bent on war. From my reading, Grey's musings about resignation are based on principles of conscience/ honor. At the end of the day, he can stay or go - if he resigns, he loses the ability to influence future events, which is why I believe he would ultimately suck it up and stay.
I think it was possible (though not certain) that Grey could have caused the collapse of the Asquith government whether Asquith joined him in resigning, or not. For Asquith, if Grey resigned from a neutralist cabinet and he did not follow, it may have been political suicide.
In the first instance, how does Grey cause the collapse of the Asquith government (without Asquith) - Grey resigns and is gone. How exactly would accepting Grey's resignation be political suicide? To recap, even with the expectation of the Germans invading Belgium, only 2-3 British members of the Liberal Cabinet were enthusiastic on intervention.
Yes, I think Asquith's motive in both cases was to suggest neutrality was possible when it was not, was the personal embarrassment at the fact that Britain must come in to a Franco-German war and nothing Germany did or did not do could change the fact. This was a the heart, the problem with the Entente policy. It was sold as defensive in nature,, but given the danger of German hegemony, whether Germany or France started the war was immaterial to the fact Britain must join it. That, IMO, was the source of Asquith's embarrassment in the days before Germany invaded Belgium.
This is one of those theories you have, which is not supported by the available facts. By most accounts Asquith attempted to lead by consensus and his delegation of duties having what can only be described as loose oversight. I don't think there is any reputable historian who has suggested Britain was compelled to protect France (actually Russia) in almost all circumstances.
Asquith had a number of priorities including, keeping the Liberal party together (and in power), navigating the Irish rule matter, maintaining the peace in Europe (if possible), defending Belgium territory against aggression, supporting his Foreign Secretary (and thereby France), while seeking not the alienate Germany. Prioritising the theoretical threat of German European hegemony over pressing current matters, while also discounting matters pertaining to the future Franco-Russian alliance and avoiding maintaining any written record of this thinking is rather impressive.
The embarrassment and anxiety that Asquith felt for the situation at the time afterwards has been matched by endless historians and students of history since, thousands and millions of which have decided that Germany's actions after 31 July had any material impact on the British entry into the war, when it did not. Myself, I understand Grey's policy perfectly and see why it had to be that way. The Germans could have avoided war with Britain, but not by 31 July. Just like how Grey had warned the Germans on 29 July.
Clearly those historians have been needlessly shackled to the facts - and are not free thinkers like yourself. I note you paint the British PM as the decider, but then choose to prioritise Grey's position when it suits you and then claim Asquith lied when it does not. Very liberated and imaginative thinking - very difficult to argue with!