They said "most likely"? They couldn't tell you for CERTAIN? If so, then they didn't test anything. I sure hope you didn't pay them anything to tell you it could be the PSU, motherboard or CPU, all of which we already knew it COULD be. The whole point was to have them tell you what it was or was not.
If you bought new memory, then it wasn't that.
An RX460 WITH the rest of the system, only needs a decent 400w power supply, so that 430w Smart PSU that you said was new, should have been perfectly fine for that configuration. If it wasn't new, then sure, it could be the problem. If it was new, then it most likely is not. There is absolutely no reason why you should need a 650 or 750w power supply for that configuration. None.
If they had TESTED the graphics card, which is what they were supposed to do, we'd know one way or the other whether it was or was not the graphics card. But since they clearly did not, that was simply a waste of time and hopefully was not also a waste of your money. If they charged you ANYTHING to tell you "well, it might be this or that" then you should go back and demand a refund. Perhaps even threaten to get the local District Attorney and consumer affairs involved or report them to the regional branch of the Federal trade commision (FTC) that oversees consumer affairs. If you are in the US anyhow. If not, then whatever agencies handle such things in your country or region. If they did not charge you anything, then going to a more REPUTABLE repair shop that CAN test the card in another system to verify it is good or bad would still be the first thing you should do.
Guessing what is wrong is not what you want to be doing. Without knowing if the graphics card even works there is no way of knowing if anything else is not working. We need to KNOW that the GPU is either good or not good. If it is not good, then you need to get one that works. If it IS good, then we need to figure out why there is no display and what is causing it.