A 5lb cylinder contains as much CO2 as 25*88g cartridges (you linked the 20g system, which is less than 1/4th the size). How much do 25 of those cartridges cost? And, you either turn it on and off each day, or you leave it on 24/7. If you leave it on, you replace cartridges 3 times as fast, meaning it'll cost you 75 88g cartridges - or 340 20g cartridges - with that system to run as long as a 5lb clyinder (with regulator and solenoid).
On a nano tank, the cost over time is much slower to increase, and since nanos are often on a table or counter, there's no good place to hide the cylinder. And for a 5g tank, a 5lb cylinder could conceivably last 3 years, so a bit overkill.
Yes, that system will provide enough CO2 for any plants, carpet or otherwise, given sufficient light and nutrients. If (on a 5g tank) you turn it off and on every day, an 88g cartridge could last a month and a half. Full time, you'll replace every 2 weeks.
You can't use a solenoid for this thing because there's no regulator. The pressure in the line would build up to 800-ish psi against a blockage, and solenoids we use in this hobby are designed for (usually) a bit over 100 psi. And if you *do* turn it on and off every day, every day you'll have to tweak it to the right flow, and you risk killing off your betta, or shrimp, or microrasboras if you go too high.
An entry-level pressurized CO2 system with cylinder, regulator, solenoid, diffuser, and any extras will cost you in the $150-200 range. Unfortunately, CO2 ain't cheap to start. But the only cost over time is refills.