Elevated Deck over concrete Patio
Since you’re not making your concrete patio as deep as the footers, your patio will be subject to frost heaving. This means that if you mechanically connect (e.g. build in a single pour) footers and patio, you’ll make your footers subject to frost heaving as well, even though they reach below the frost line.

I believe heaving of the patio might be mitigated if you pour it over a bed of crushed stone, like basement and garage slabs are done, but I don’t have experience with this.

Side note: you should use concrete tube forms regardless, as they will make your footers relatively smooth, making it much harder for anything subject to heaving to “catch” them and jack them up.