I'm in a similar boat- wanting to use original hardware, but lacking the space for a CRT to display it properly.
You've got 2 options here- convert the video, and upscale the video.
Upscale will alter the video signal to be higher resolution so it looks much nicer on your HDTV, better than CRT depending on who you ask. Trick is, it's very expensive to do right- we're talking several hundred dollars here. The $100 unit your friend linked to seems to be a low-end upscaler. This is a bad idea- assuming it'll put out a decent signal, you don't know what kind of input lag it will introduce into your setup. See, it takes time for the video signal to get altered and displayed. For movies, this isn't an issue, but for games where you're attempting to manipulate something within the video? You can get scenarios where the character doesn't move right away after you tell them to, and really screw up your gameplay. If you're not gonna pony up for a good setup, it's not worth doing.
A Converter will simply change the video signal to something HDMI can understand, without trying to improve the resolution. As a result, the picture is worse, but the units cost less. That's the $15 dollar thing you found. The problem with converters is, if they're poorly made, you can get especially bad video quality, poor color/saturation, and even still run into lag problems. Now, for various space & cost reasons, I use a converter. I've gone through a couple now, but my current unit by enKo is excellent. I'd give you the Amazon link, but for some reason it's not currently available. My tips:
-Don't buy the unit from Musou that Amazon advertises. The colors are extremely muddied, and mine died 1 year in.
-Check the inputs before you buy. They do make units that convert HDMI into composite so you can use newer equipment on old displays, you don't want to get one by mistake.
-Be ready to return/replace anything you buy. A lot of these are cheap Chinese junk where your performance can vary wildly even within the same production run. Make sure you can deal with getting a dud.
I've been using this: https://smile.amazon.com/gp/product/B00V2ULHBS/
It's basic, and everything is still stretched out, but I haven't been able to detect any lag. Supports 720 and 1080.
I've only had it for a few weeks, though, so this is not an endorsement. I primarily use a CRT.
One last tip for both of you: find the video settings for whichever input your converter is plugged into, and CHANGE YOUR RATIO TO 4:3. It fixes the stretch & improves the image. Very important step if you're using a converter!