I bought the Jak and Daxter collection for ps4 and have been quite disappointed at how much it lags. Do PS2 games released on the PS4 Pro make use of the better hardware and run at a more consistent frame rate?
Generally speaking, no.
This is going to be more of a general answer than a specific one for PS2 games. A good example of this would be Red Alert 2 for the PC. I say no because, although you may be playing it on a modern system, which is orders of magnitude more powerful that the original it was developed for, the actual implementation of the code may not be able to actually use it.
A simple way of thinking about this would be, you are trying to plug in as many power cables as possible into a home appliance. The more cables, the faster it runs(the game). These cables are hooked up directly to a generator, ie, your console. If the console runs out of power slots, it cant make the appliance run any faster. The reverse is also true. If the appliance itself was designed for a generator with three plugs, then it will more than likely only have three plugs. So although you may have a console with 25 slots, only three can be used. This is a simple way of thinking of multi-core processing and hyper-threading, etc. This is why when nukes are used in Red Alert, the game lags, not because the console isn't powerful enough, but because the game itself is not capable of using the consoles full potential.
A hilarious inversion of this is in very old games, where the game physics itself is tied directly into the speed of your CPU. So for games like the original XCOM, a game turn which would normally take a few seconds on a 1990's computer would happen in a split second on a brand new computer and you wouldn't even really see what happened.
As mentioned above, emulation is also something to consider. When you emulate a game, you are doing so because the game will not run natively on your console. Meaning, if you are on your PS4 and it has to emulate the PS2 to play the game you want, that means it is running both the PS4's operating system, and the PS2's. So not only do you end up having more overhead, you also run the risk of emulation errors happening, as emulation is not a perfect simulation of the original console. You generally do not have to worry about the overhead for very old games, as the difference in requirements is pretty extreme, but it is a consideration when you are playing games that are only one generation apart. So if you are playing Xbox games on a 360, or PS3 games on a PS4, emulation overhead would be more of a concern.
However, if the game is re-released for the PS4, it essentially isn't a PS2 game anymore.
(Source: Software Engineer and Gamer)