To be fair didn’t say it to replace it completely , he said use it for a while and you will be stronger at the barbells. He might be , I picked one up for 2 pound at a Charity Shop (less than five dollars.) I used it twice so maybe I not in a great position to comment.
That said it could certainly give you some work as difficult as barbells especially in the lower back and for the repetition method. Pike pushups on it i.e. handstand pushups certainly especially if add a weighted vest and perhaps use pushup bars to increase range of motion.
I would defintely keep deadlifts or squats in every 2 weeks or so or more often if you do lose strength.
The exercise which approximates the standing ab wheel is very hard and good too. Ab wheel is great for lower back strength too. Back extension including weighted are great , also glute ham raise.
There is an exercise you can do called hamstring curls with the TRX - heels on works hamstrings, toes through feels like the cramping associated with glute ham raises but obviously a far better easier. You can use a small ladder and platform to make more difficult.
Hip pushups to train for planche pushups are good too. I have been trying that so far. Not sure how it will have affected my barbell strength.
I can’t recommend the baoding balls(chinese chiming balls which you rotate) for grip strength and hand recovery more highly. I got some boules balls which weighed 630g each at a car boot today and could rotate them with difficulty. Apparently according to John Brookfield you should build up to shot puts in each hand.
Sometimes it is easier to overdo the weights so much that they affect your skill sesssions and that is perhaps what your trainer might be getting at .
Barbells are great but they do beat the shit out of your joints over time, dumbbells are perhaps better in the long but a pain in the ass to set up and expensive.
Presently I train with the TRX (recovering really) but it seems to train my lower back in a safe way .
Slow very controlled pushups are probably one of the best core exercises ever. Almost like a bodybuilder would do them . They are called high tension pushups I think (described also Tsatsouline’s books) . I had quite a sadistic Judo Instructor who used to prescribe these. I never believed in them , I do so now .
I would imagine the heavy cheat rows must be amazing for experienced grappers. He also prescribed chinups hanging from a judo gi. He was quite small 5, 4 and about 200lb and one of the strongest man I have met and could easily outjump me too for standing broadjump when I was jumping around 2.30/2.40m.
The TRX is very good for circuits too. For example you can pushups then immediately flip your feet to do hamstring curls.
Or if got a platform for make the rows difficult enough you could those too.
To summarise I think you could completely replace barbell work with a TRX and some bodyweigh work , chuck a weighted vest it here and there. Add bw plyos and you would be golden , also chinups too and back extensions and glute-ham raises.
I also dismissed it as useless before using. That said though it is particularly easier to use or adjust and I wouldn’t have got it if I paid anything like full price for it.
Also underestimated are pushups bars for slow controlled perfect form pushups. Obviously throw some faster ones in here and there .
A sled is good work too with handles .Chest expanders would probably work well or safer now heavy duty bands , which can use for repetition lower body work.