@makauchan most capture cards handle all the heavy lifting on your own and pretty much any PC made in the last decade should be able to handle a capture card. PS4 remote play would both need to remotely connect and play the games while recording the game. I'd personally recommend getting a capture card (the one I use is the mirabox, but you can find other stuff here: https://www.speedrun.com/streaming_recording_equipment/thread/1mkl1).
That being said, it could just be that you have extremely high/bad recording settings. If you wanna try and get remote play working though it'd be nice if you could answer some questions. 1.) What are your computer's specs? 2.) Can you showcase what the stuttering looks like? 3.) Assuming you're using OBS, what are your OBS settings (stuff like high bitrate/resolution/the type of capture can have drastic effects on FPS)?
Also, just pointing a phone or camera at your TV is always the cheapest and easiest option as Pear said, so if you don't wanna spend money and don't particularly care about the quality of your run I'd recommend doing that.