My thoughts - it is not a training method I would use but if you are happy with your trainer and how your puppy is doing that is the important bit.
I personally would not use a harness on any dog all dog, especially a fine skinned dog such as an Italian Greyhound, who looks very sweet in the photo - a lovely breed.
I would search for sighthound house collars and use one of those, with the caveat of always making sure the puppy cannot get caught up on anything with the collar on. Connect a lead to the collar when needed, however also be aware, especially for such a delicate breed as an IG, it is easy for their necks/spines to be damaged if they pull or you go in opposite directions.
My opinions are: a harness is better than a collar but not on all day, I would find a harness which just has a chest strap and one buckle/fitting - it is then just placed over the dogs head and done up, takes seconds.
If that is not an option a soft house collar, but I also would not leave it on the puppy all the time.
Finally a soft leather half check collar could be used, easy to put on, in a second and clip a lead to it.
Irrespective of training method used - the types of leads, collars and harnesses used can indicate to dogs and therefore train them to know what is expected of them, so leaving them on all the time defeats the training methods. That may not be clear, so as an example - dogs which do cani-cross know when their harness goes on, they will do cani-cross, when a show dog has a very fine show lead on, they know they are going to show. When a dog has its "normal" lead or harness on it knows it is just going for a walk.
Good luck with your IG, a lovely breed.