There is no American Goldfinch in this clip- as I stated earlier the two species we can clearly hear are House Finch and Curve-billed Thrasher. American Goldfinch is actually quite rare throughout most of Arizona. The first few notes of the House Finch song do sound a bit 'goldfinch-like', but in context it's definitely a House Finch.
As to the thrasher, remember that it has a diverse set of calls and songs- you are probably familiar with the classic whistled 'whip-whip' call as it's a common sound in suburban AZ. Their songs however, are a series of (often) twice-repeated phrases. Simply based on the fact that you mentioned you're in Sierra Vista (as opposed to far outside the city) it's pretty safe to call it Curve-billed- Crissal, Bendire's, and Sage Thrasher are not really 'suburban' birds. On top of that, the quality is exactly what I'd expect from Curve-billed (lacking the huskier, more musical quality of Crissal), and Sage Thrasher or Bendire's would be exceptional for that location.