I have heard that getting separates is generally better all around
Well...
...no.
I would not say, today, that separates are generally better than an integrated. There are tradeoffs involved both ways. Expense is the biggest, as you mention. Size is another. Actually, at a given price level, I think you should in principle get better sound from an integrated than from separates. Power supply structure and shielding need to be good in the integrated but you eliminate a set of interconnects, and money saved on the extra box can be spent on better parts.