The central message from the likes of BE is that there is no viable domestic market to LHR, in part because of competition from cheaper and/or better alternate options. For connecting passengers this is the European hubs, for O&D it is the other London area airport's or the railways.
You are doing what many of those against LHR expansion have been doing. You are trying to argue the case based on the current setup not the proposed final outcome. A third runway giving access onto LH routes would provide that very need for the domestic traffic from the regionals.
I don't agree that the people of Yorkshire and other regions don't care whether they change at LHR or AMS etc. When looking at connections there is always the concern that you will get stuck and it is seen as far better to be stuck in London than over the water.
Also, what are these environmental constraints you talk about?
As far as i'm concerned this will not be agreed until a decision has been made as to whether the UK wants a hub or not. If it is agreed that we do want to capture some of that traffic and income that currently goes through the likes of AMS then LHR expansion is the only answer.
HS4 will connect Heathrow and Gatwick. The closer the approval process for this gets the less chance there is of a third runway. connecting between the two via a rail line however will not be a good option however much they try and sell it.
So we really need a separate debate about the benefits to the UK of a hub setup. Only when we have answered that should we move on to whether we should build it.
Personally I believe that in a post Brexit world it will be a major asset. I also think it will provide much better access from the regonals to LH destinations as well as giving the south east more viable destinations.