Part of the reason to use the Red Cross (or any NGO) is because it can do things outside the full government purview.

It's also a good way to get money from citizens across the country to assist the people affected. Think of it as AB spending $1 to get $1 from the feds and $1 from citizens/donations. Imagine how people would react if the AB government put out a request for donations directly.

it's the f-ing government - either just go help people who need help because they're your people or contract with red cross to do the helping for you with accountability for the job they do.

Agreed, and I believe that both of these are happening in Alberta. The Red Cross is contracted by the AB and local governments to provide certain services in disaster management, and the AB and local governments provide some of the services themselves.

I don't feel like the government should be matching donations to a private organization

It's an NGO, not-for-profit, registered charity, with a mandate to provide humanitarian aid. Calling it a "private org" is a bit disingenuous.

From the Red Cross:

Donations to the Canadian Red Cross will be used for assisting those impacted in Jasper and other affected areas in Alberta with immediate and ongoing relief including financial assistance, support to evacuees and the communities hosting them, recovery and resilience efforts in response to the wildfires, as well as supporting community preparedness and risk reduction for future all-hazard disaster events within Alberta.

I don't know how (or if) AB audits this, which is the crux of your issue with it I guess, but in a way they don't have to. If the Red Cross provides services for Albertans in a manner that is close to the value AB spent, they haven't lost anything. Hopefully it's (much) better than break even though.

And before we get into the efficiency of the Red Cross for dollars donated, remember we're talking about governments here. They're not exactly the paragons of efficient spending.