The core problem is that there isn't one Conservative Party, there are two and they don't agree on the central issue of the day. One faction joined the party because it wanted to do what was best for business and doesn't really worry too much about ideology so long as their party can create a set of circumstances that enable businesses and their owners to prosper. That faction is particularly worried about what will happen if we leave the European Single Market. The strength of this faction is why the party campaigned to remain and why the majority of their MPs were happy to do so.
The second faction is driven by ideology. They think the EU is a collectivist plot and the world can't ever be at rights if the state is doing anything beyond building up the armed forces and operating a robust police force and law enforcement system. They basically believe the same ideology as the Tea Party or Donald Trump and they want out of everything to do with the EU including the single market because they think all of that is a moral outrage. Since the Brexit vote they have been in the ascendancy.
Practical business people and ideologically driven believers in their own propaganda rarely get on very well. They managed to do so after May became Prime Minster by virtue of one single thing. They both saw the advantages of being in power. They both thought their best chance of implementing their objectives was to avoid public argument and to get behind a strong and stable leader. After all, she hadn't yet committed to any actual policies with regard to Brexit.
May now faces two enormous challenges. The first is that she now does have to start negotiating Brexit and it is impossible to do so without your own party staring to find out what your actual policies are. As soon as she commits to any policy on any aspect of Brexit one faction or the other will hate it. This also applies to UK policy post Brexit. The Tories don't agree with themselves on issues as widespread as the role of the state in regional policy or whether agricultural subsidies are a good idea. They don't agree about whether we should be in or out of the single market. They don't even agree about controlling free movement of labour since the business faction wants pretty free immigration and the ideologues want next to none.
Strangely this factionalism is the main reason I think May could survive. She is damaged goods but there is not much else available in the shop that the whole of the party wants to buy. Osborne has gone and is enjoying criticising from the sidelines every move May makes. Gove ruined himself with clumsy over ambition. Johnson has proved incapable of serious government responsibility and would be every bit as easy meat for Labour as he always was for a couple of comedians on "Have I Got News for You". The other alternatives almost all carry their own damage or have such low brand recognition that the public would say "Who?!".
Imagine the situation. A new Conservative leader emerges from a pack of obscure squabbling contenders after a bitter battle. She stands before us (in my opinion the most credible remaining contenders are female) and tells us she is going to lead our country forward. First question please! "Tell me Prime Minister, when are you going to call an election to legitimise your authority - because no one voted for you to be our leader?"
What new Conservative leader would be brave enough to lead a divided squabbling party into an election against a Labour Party that has finally come together behind Corbyn? What new Conservative Party leader would be brave enough to face all those newly registered young voters who actually turn up and kick you out in Kensington and Chelsea and Cantebury? What new leader could cope week after week with the accusation from a newly confident opposition that she was running scared of holding an election.
So the Tories will have to think long and hard before they replace May. She has led them into a position of weakness but at least they are in power. Any attempt to replace her could see open warfare break out and lead them into an election where they are really badly beaten. There will be weeks of rumbling and serious attempts to find a unity candidate to crown but I think they will fail to agree on the choice of replacement, fail to find a candidate of quality prepared to risk all on the challenge and fail to take the heavy gamble that a new replacement will be able to survive and prosper.
I therefore think we face a period of weak administration run by a Prime Minister who doesn't like unpredictable change, doesn't do delegation, wants to think long and carefully before she makes a decision and doesn't want to consult many people about it or explain it to any actual voters.
This might not have been altogether a bad thing as it might have left us with a pragmatic compromise of a cobbled together Brexit instead of an ideologically pure clean break. What makes it really bad is the deal with the DUP. Socially this mob are about as aggressively unenlightened as you are ever likely to encounter. We are now about to have the far right tail of the DUP wagging the policy of the whole of the UK. In case you missed it you might like to note that the DUP got 292,317 votes. The Greens got almost twice as many on 525,371. How can anyone justify a party as small and as extreme as the DUP making policy for the entire United Kingdom? Even I don't think the Greens very much bigger share of its vote should entitle us to set the national agenda!
I am not the only person who ins't impressed by the DUP's policies. May's majority was actually produced by the Scottish Conservatives winning seats from the SNP. The woman who did that is Ruth Donaldson. She is about to enter into a same sex marriage. How is a woman like her supposed to cope with being in government with people who thinks she is a sinner and her marriage is an abomination!
I wish Theresa May the best of luck in working out an answer to that question!