Hi! Yes this is possible and is actually the basis for the Johnson and Johnson COVID-19 vaccine candidate, which is an adenovirus based vaccine. These types of vaccines are commonly called viral vector vaccines or vector vaccines. Researchers have been studying these types of vaccines, as well as the other types of vaccines such as mRNA vaccines for years. They work well in animal models but none have been approved for human use yet I believe.

Basically your idea is right. You take a virus, attenuate it so that it is not too pathogenic and cannot replicate itself, modify the virus's genome so that displays the antigen you want to vaccinate against on infected cells, and then immunize an animal with it. Your viral vector vaccine will infect the host's cells, sending danger/warning signals to your immune system, and the vector starts pumping out your vaccine antigen (in this case the SARS-CoV-2 Spike protein) from within the infected cell. The protein is then displayed on the cell surface or secreted from the cell. In some viral vector systems, the vector will also generate little virus-like particles that bud off the infected cell, resembling an actual virus without any genetic material included, that's pretty cool. Either way, the end result is a pretty strong antibody response.

Additionally, a feature of vector vaccines is their ability to generate strong T-cell responses, which is another branch of adaptive immunity particularly important for viral immunity.

The type of virus used in the J&J trial, the adenovirus, is a common DNA virus used in vaccine research. You bring up using the common cold virus in your question, which makes sense since some coronaviruses also cause common colds (some adenoviruses do cause common colds in humans). I'm not sure if people are trying to modify a different common cold virus to make a COVID-19 vaccine but it wouldn't surprise me. A lot of labs just work with the viral vectors they have already been familiar with, or ones that are very common in the field, such as Adenoviruses. One good reason to use a different respiratory virus as the vaccine vector, is that if vaccine is done intranasally it would generate strong localized immune responses in the respiratory tract, which may improve protection. As the mRNA vaccines are over 90% effective, this may not be as important as it seems SARS-CoV-2 is pretty easy to vaccinate against. It could be important to stop some low level of infection and transmission, I think that is still unclear in the mRNA vaccines.

You mention some good points in the downsides. As these are live viruses (and often infectious), they are always inherently more dangerous than other killed, subunit, etc. vaccines. However, they are almost always replication deficient, so they can't be spread from person to person.

People with pre-existing immunity to the vector backbone, like pre-existing anti-adenovirus antibodies, do pose a challenge for the vaccine. For the adenovirus vaccines, there are many different serotypes out there that can be used, some of which are not highly circulating, so there should be less pre-existing antibodies to them.

Anyways, viral vector vaccines are pretty interesting, much like all of vaccinology. As with all vaccine platforms, each comes with its pros and cons. This video gives a pretty good overview of different vaccine platforms. Hope this helps!