Bonjour! (I won't pretend to know more French than that, hehehe)
Welcome.
Your all-in-one system breaking down is actually a blessing in disguise. Those tend to have very heavy weight on the tone arm, which causes your records to wear out faster. Not only that, but they don't sound that great.
For an affordable but good-sounding turntable, check out the Audio Technica AT60 ($100) or AT120. ($250)
The AT60 is automatic, 33 or 45 RPM, internal or bypass pre-amp, and belt-driven.
The AT120 is manual, 33, 45, or 78 RPM, internal or bypass pre-amp, and direct-drive. They both have a USB output and software included, so you can rip your records to MP3 or whatever.
If money is tight and you don't have a proper amplifier or receiver/amplifier, you can hook up the output of the above turntables to amplified computer speakers, and it will sound decent. (probably better than your all-in-one) If you don't have them yet, get some with a subwoofer, so you don't miss out on all the low frequencies.
Later, when you get a proper amp and speakers for the turntable, the computer speakers can go with a computer.
I just re-discovered vinyl here myself. The records are sometimes given away or sold for very cheap here, so the problem is storage space and trying to make myself selective enough not to take anything.
You said you're into hard rock. It reminds me that I need to get some Scorpions records for my collection...
Again, Welcome!