If the Secretary of the Treasury in 2008 hadn't been from Goldman Sachs, the collapse could have played out quite differently. Bush was inclined to let Wall Street go bust. Goldman was about a month from bankruptcy when the bailouts started.
If there hadn't been a bailout, the result probably would have been more like the 1980s S&L crisis, with the FDIC and FSLIC taking over failed institutions, winding them down, and selling off the real estate. Non-bank institutions like Goldman (which wasn't a bank at the time) would have just gone bust, leaving their investors with zero.
It's a good question whether Henry Paulson showed favoritism to Goldman Sachs, but we need evidence to make a judgment about the answer. The rest is speculation about a hypothetical person's choices and their outcomes in a highly complex, chaotic, novel situation.
Most experts, especially those within the establishment, say that allowing all those institutions to collapse would have been catastrophic and that the government should have rescued them. I think it's reasonable to expect that if one of those establishment experts was the alternative to Paulson, a likely scenario, then they would have acted similarly.
If someone acts to prevent catastrophe and succeeds, critics always can (and do) say that the catastrophe wouldn't have happened anyway.
Paulson's brother worked for Lehman Brothers, and it doesn't take a lot of reading between the lines to see that this was a strong factor in why the company did not receive any support.
In contrast, concerns about nepotism didn't apply to the relationship between the Treasury and other banks, although the Treasury seemed sensitive to Goldman (JP Morgan and Jamie Dimon got better treatment). Personally, I think the Treasury and the Fed did a good job in response to the crisis, and that Congress abdicated almost all responsibility for dealing with the crisis, leaving the financial response dominated by monetary policy (which by definition favors the financial sector by promoting unequal access to terms of credit) instead of a more egalitarian process.
If there hadn't been a bailout, the result probably would have been more like the 1980s S&L crisis, with the FDIC and FSLIC taking over failed institutions, winding them down, and selling off the real estate. Non-bank institutions like Goldman (which wasn't a bank at the time) would have just gone bust, leaving their investors with zero.