In 2010, much of Greece government's debt was in the hand of private investors, especially French and German banks and there were serious concerns that a default could trigger a contagion that would destabilize the whole euro area. So even before any talk of leaving the euro, simply defaulting on the debt could have serious consequences. It might not have been the only explanation but that's a major reason for everyone's attitude back then.
On the other hand, Germany (and some other countries like Finland) had a ready-made medicine to offer: Greece just needed to be more like them, spend less, conduct “structural reforms”, regain competitiveness by depressing wages, no need to leave the euro and all would be well.
In reality, wage moderation is useless if you don't have favorable conditions (namely a decent rate of inflation and growing economy everywhere else) like Germany had in the 1990s and austerity is counter-productive when everybody does it at the same time but, because Greece was clearly spending too much relative to its tax revenue and even cheated to hide it from European institutions, it was still kind of possible to pretend that budget deficit was the only problem and that reducing public spending would solve everything.
Five years on, the debt is in public hands, with zero risks to the banking system outside Greece and many believe Greece has been “ring-fenced” and the Eurozone would not suffer too much if the country were to default and/or leave the euro. Although the latter aspect is disputed and long-term consequences are uncertain, that's enough for several leaders to be more comfortable contemplating an exit now.
Furthermore, on the second point, the austerity programme has been an unmitigated disaster, with an emerging consensus among economists all the way to the IMF that austerity was a very bad idea in current conditions. Spain, Portugal and Ireland haven't recovered from the crisis and only suffered less than Greece because austerity was less intense there. Even countries that do not fit the “spendthrift Southerner” cliché like Finland or the Netherlands are not doing well and only look good because the rest are even worse.
That might seem incidental to your question but it does matter because it means it's just not possible to argue that Greece has a bright future in this euro area, if only it would do what it's asked. Some people in Northern Europe still like to imply as much but it's certainly not convincing to the Greek people who know first-hand how much their country has already done and obviously cannot buy the fantasy that the only problem is handouts and excessive spending.
At this point, avoiding an exit from the Eurozone requires an agreement with the creditor countries and the IMF that satisfies the ECB or some other way to protect the Greek banking system. The Greek government is clearly ready to sacrifice a lot to that end but it cannot simply accept another round of senseless austerity with no debt restructuring, no concession from the other side and no resolution in sight.
By contrast, for the German government (and many other people) accepting anything else than more of the same would call the whole approach into question and belie the narrative they have been peddling to their own citizens for years. So, sincerely or not, they prefer to continue pretending that Greece “needs to do its home work” and risk a Greek exit rather than share the blame and question the central tenets of their economic policy.
From that perspective, pushing Greece out – just like the continuing focus on Greece budget situation prior to the crisis, on the tone of the negotiations or on what the current government did or did not do – is nothing more than a way to avoid facing the fact that austerity is an integral part of the problem. The irony is that while is politically expedient for many people in Europe and without speaking about any potential long-term effects, it does not even protect the creditors' narrow interests as it would mean that they will recover even less money than if they agreed to the debt write-off they so adamantly reject.