Interesting question - but there's an important distinction that needs to be made. Once you trace and expand an image - as you describe - it's no longer an 'image' in the photographic/pixel-based sense. It has been converted to a vector-based representation.
With that in mind, one solution to achieve this (without dividing the text into lots of pieces), is to use a pathfinder operation. Here's how;
1. With your text in front of your object/image, open WINDOW > PATHFINDER. Make sure both items are selected and ALT-click (or OPTION-click, if using a Mac), on the fourth Shape Mode, 'Exclude'.
Visually, this will look largely identical to the example you've presented in your original post. However, because you've ALT/OPTION-clicked, this will create a compound shape - and the text can be moved about, edited and otherwise manipulated by simply drilling down to it in the layers panel. The text has NOT been divided into any pieces, so it's extremely convenient for this kind of purpose.
If you had instead converted your text to paths and used the regular Exclude pathfinder operation (without ALT/OPTION-Clicking), you would have nothing like this kind of flexibility.
Also note that the fill and stroke attributes are applied throughout your newly-created compound shape from the front-most item (in this case, the text), which is probably what you've already noticed.
2a. Now, let's consider the same issue with an expanded image. I don't have your bird logo or texture, so I'll just use something else.
Here, I have a text layer, as before, and an image that has been traced, expanded, united and converted into a compound path using Exclude. Uniting and converting to a compound path is not strictly necessary - the group of paths that resulted from the image trace would have worked equally as well, but I prefer to keep things tidy. Also, it may be necessary to remove any white or redundant fills from your group before applying either pathfinder operation, or results may not be as expected.
2b. As before, with your text in front, select both your text and your compound path or group. ALT/OPTION-Click on Unite to create a compound shape. You should see something like this;
2c. Now, in the example you've used, you appear to have used a texture instead of a vector-based pattern or fill. We can simply apply a pattern swatch to the fill of our new compound shape to do this.
Here's how: place your pixel-based image into your project and drag it into your swatches panel in order to create a pattern swatch. You can now delete or hide the placed image if you wish.
2d. With your compound shape selected and the fill color active (otherwise you'll create and apply this to a stroke ...), click on your new pattern swatch to apply it to your compound shape;
2e. You'll notice that, in my case, even though the texture is large enough to completely cover my compound shape, it repeats toward the bottom, creating an ugly break. This is because my pattern swatch has begun its repetition at the top of my artboard - it's not centered within my compound shape.
You can fix this by selecting your compound shape, navigating to WINDOW > APPEARANCE and selecting the patterned fill. Now click on EFFECT > DISTORT & TRANSFORM > TRANSFORM to bring up the 'Transform Effect' window.
With the 'Preview' box checked (to make your changes easier to see), make sure that the 'Transform Patterns' is selected - but 'Transform Objects' is UNchecked (otherwise you'll move the entire compound shape, which is no good for what we're trying to do).
You can now use the Move > Vertical section in order to shift the placement of your pattern fill (or horizontal, or scale, if you wish).
In this way, you'll be able to use a pixel-based pattern/image on a traced and expanded background whilst retaining fully editable text.
Hope that helps.