A 2.8 becomes a 4.48 when you factor in the crop on a canon which is 1.6
I'd be VERY careful how you phrase some statements to avoid causing confusion.
The aperture in terms of exposure is the same no matter the camera body behind a lens. f2.8 is always f2.8 in exposure terms. This is a proven and well known bit of info that we all know without realising - because external light meters do work for any format (sensor) size and yet never require the sensor/film size to be entered in.
Now in terms of depth of field you are correct, the difference in depth of field between a 1.5 or 1.6 crop camera and a 35mm is roughly similar to around 1 stop in aperture. The full frame camera will have the lesser amount of depth of field.
An extreme way to also show this is if you use a mobile phone camera, where the sensor is tiny. In those you can easily see that the photos often have very large depth of field; indeed its often very hard to get background blurring with such a tiny sensor.
The other end of the extreme is medium format cameras which offer far more background blurring effect, again because the depth of field for the larger sensor/film is so much greater.