I grew up with the saying "There's no replacement for displacement" when referring to engine size, performance, etc. Does that saying still hold true today?
First of all, you're kind of mixing apples and oranges. Here's why ...
The saying of "There's no replacement for displacement" is not about reliability (as the 2nd part of your question is asking), it's purely about torque and power. What was true in the 1950's & '60's still hold true today. The reason is, when you look at any comparable engine of different sizes, the torque output is much greater for the larger engine, all other things being equal.
Let's use the LSx platform since I know more about them. Look at two different engine sizes of the LS platform, say the 4.8L V8 (LR4) and the 6.0L V8 (LQ4) from a 2004 Chevrolet Silverado. The truck versions of these engines were comparable in make and composition. The heads were comparable, while of course the cam would be slightly larger in the 6.0L to take advantage of the displacement. They even had nearly the same compression ratio. (Note: Other than displacement, these other three areas are main parts of what can change power on an engine ... for the most part. There's a lot of other little things, too, but that would be getting too far a field for this answer.) Even so, the hp/tq output of these two engines were drastically different. The LR4 had ratings of 285 hp / 295 lb-ft, while the LQ4 had 325 hp / 365 lb-ft. That 70 lb-ft of difference was why GM put these higher output engines in their 3/4 ton trucks (2500 & 2500HD). The extra torque would allow them to tow a lot more weight. Here is a prime example why displacement is king.
... would a 4-banger pushing 300 hp last as long as a 6 or 8 cyl pushing the same? Can the smaller rods, cams, etc. stand up to high torque output and maintain their reliability?
This part of your question deals with longevity and reliability. So it is truly a different question indeed.
When looking at a 4-cyl and an 8-cyl with the same horsepower/torque ratings (you really wouldn't find it in the real world, but we'll go with it here on the premise you could), if all things were equal other than the displacement, the 8-cyl engine would most likely easily outlast the 4-cyl engine. The reason for this is, the 8-cyl engine would be able to make the hp/tq numbers way lower in the RPM range than would the 4-cyl. Being able to do this, there is far less stress put onto the 8-cyl as compared to the 4-cyl. Many years ago a man by the name of Carroll Smith wrote a book called, Engineer to Win (Note: Edition I have is from 1984). In this book, Mr. Smith writes about how metal has a life span to it. It can go through "so many" cycles before it gives up. It doesn't matter if it's a connecting rod or a valve spring. It only has so much life in it. He states, the harder you push the metal, the shorter its fatigue life. A valve spring might last a million cycles if not pushed past a certain limit, but if you go past that limit by just a fraction of an inch, it might only last 100k cycles. I'm just throwing numbers out there, but I'm trying to illustrate a point.
You may be wondering, how does this apply to the case of our two engines (4-cyl vs 8-cyl). If everything else was exactly the same in the two engines (ie: rod size, piston size, bearing journal sizes, etc), in order to make the same amount of power out of the 4-cyl, you'd have to push it a lot harder, using larger cam(s) or what have you. You'd have to spin it a lot higher in the rpm range. You'd have to make it do more. In doing so, you will reach it's fatigue limit much faster than you would the 8-cyl. Something will break sooner. Why? Because you would have to drastically stress the 4-cyl much harder than you would the 8-cyl.
Now, are larger engines today going to last longer than smaller engines in the current day and age? It all depends on how the engines are built. I don't have empirical evidence of this, but it would logically make sense if, as a modern vehicle manufacturer, you are going to sell a vehicle where you have 100k mile warranties, you're going to make the engine (under normal circumstances) be able to live for at least that amount of mileage. Or at least that'd be your goal. It doesn't matter if it's a 4-, 6-, or 8-cyl engine, you're going to put the engineering behind it to last that long, or you'd suffer the consequences of having to replace a LOT of engines.
Does that mean every engine on the road today is going to last that long? By no means. There are a lot of extraneous factors involved, number one of those is, How well does the owner take care of it? If regular maintenance is not performed as prescribed, the engine/vehicle will NOT last as long as it should, so all bets are off.
Engineering is the key to all of this. I would expect a well engineered 4-cyl engine pushing out 300hp to have the same longevity as a 300hp V8 these days. Are they equal otherwise? Absolutely not. The 4-cyl is going to be built to make the horsepower as well as built to last ... at least as long as the warranty :o)