If you are looking for the "best bang for the buck" GPU, you should keep in mind that most mid-range to lower-end graphics cards currently offer fairly mediocre value on a price-to-performance ratio. Cryptocurrency miners messed up the graphics card market early last year, resulting in prices that were around double MSRP for many months, and prices are still on their way to recovery. Cards have been plummeting in price over the last few months, with many of the highest-end models returning to not much more than the prices they should be at, but mainstream cards haven't fully recovered yet. The situation is a lot better than it was last year, or even a couple months back, but still the value isn't great, with some cards offering worse performance at their price point than what was available half a decade ago. I would say overall the prices of most cards are still around 25-30% higher than they should be.
Of course, you didn't mention how much you were considering spending on the upgrade. While there are certainly cards that will be great for running most games at 1440p with high settings around the $400+ range, you might see a decent improvement even with cards priced somewhat below that. Your GTX 960 2GB was likely already struggling with some newer games even at 1080p, and framerates at 1440p will typically be around 30% lower at the same settings, or worse still if the card lacks sufficient VRAM. I would look for a card with at least 6GB of VRAM for 1440p, and 8+GB would likely remain relevant longer in future games.
Something like an RTX 2060 might be around the minimum you would want to consider, and it's currently possible to find some 6GB models of that card for not much more than $300. The 3050 8GB offers a little more VRAM for a similar price, but it's also generally around 10-15% slower in most games, and there are 12GB 2060s for not much more. The cheapest 3060 cards are currently priced around $400, and can be around 25% faster than a base 2060. And a 3060 Ti should be around 25% faster than a 3060 (non-Ti), but those are currently priced in the $500+ range. Back to the pricing situation I mentioned before, the launch MSRPs for these 30-series cards were supposed to be $250 for the 3050, $330 for the 3060, and $400 for the 3060 Ti, but you still can't find those cards for anything close to those prices.
As for the rest of the system, sure, it's going to limit performance in some titles. Especially if you are interested in running games at high frame rates to take advantage of the screen's high refresh rate. I suspect performance would be decent enough in most titles, but frame rates may be inconsistent in some newer games that can overwhelm a processor with only four threads. The prior suggestion for picking up a used i7 might be worth considering if you don't want to spend much to replace your existing CPU, motherboard and RAM right now. You could always try your i5 first though, to see what it can do. The majority of games would probably still be "okay" with it, and at the very least, your performance shouldn't be any worse than what you were getting with your existing card at 1080p. Compared to somewhat newer processors from the last few years, an overclocked i5-2500 should perform roughly in the vicinity of an i3-9100. And an overclocked i7-3770K would likely perform more like an i3-10100, which similarly brought i3s up to 4 cores with 8 threads.