You can download Streamlabs on a mobile device and stream it there. Or you can literally just record it with your camera app. As long as the game is clearly visible you're good. You'd have to somehow integrate your splits into the video, though.
Splits aren't needed for a run, the run can be retimed based on the video. It's still good to have them, but they don't need to be on screen.
Also, you can get a REALLY cheap capture card for like $13. I have one and it works pretty well.
Elgato is not the only capture card company out there. AverMedia makes capture cards on the same level as Elgato but for cheaper. That said, they are not exactly the same. If you are just beginning your YouTube/Twitch journey I would not recommend getting a high tier Elgato card. Look around at other companies and see what you can find that fits your budget.
On another note, you can do a run or two without recording and just not put it on the leaderboard. I've improved my time by about 30 mins (1:46 to 1:18) and I've never recorded a single run.
My last couple of pbs haven't been recorded. Although it's nice to submit your runs, I don't think its really necessary until top 500. Regardless, if you have a camcorder or a good phone, that works too.