You might lean out in a surplus, but more then likly you will keep it on. Once you build more muscle it will be eaiser to shred because fat to muscle ratio will change and easier to shred to lean muscle.
Regarding your inital question stop worrying about cardio and worry about calorie surplus or deficit. No its not the only way. You can go to other weight lifting forums and ask around and there are lots of ripped bodybuilders who don't do cardio. Weights and diet! (a great number of bodybuilders hate running) When shredding yes cardio (short cardio) or HIIT training are great ways to speed things up, plus its good for you to be condtioned. So yes that trainer is wrong!
When bulking Scott has taught try not to burn any extra calories while at the gym you need them to build muscle. Any extra work, play, cardio needs to be replaced while bulking. Like this morning I played football over an hour according to a website calculator and the amount of sprints I burned 800 calories. Im bulking so breakfest calories was a freebee to replace football and now I need to make sure I hit the rest of my 2500 calories today to keep muscle growing.
Again this is from all what Ive learned from other trainers and people experince.