7.0.1 will not work on an LC III or any other machine that requires an enabler. You will need System 7.1 (which you have to buy, it's not free) or later (if you want a free system that works without an enabler, Apple provides 7.5.3 as a download on their website; given the power of the LC III you'd probably be happier with it as an OS for productivity anyways).
Enablers existed to prevent double decimal point updates. Apple released a double decimal version of System 6 every time a new Mac was introduced (6.0.3 with the SE/30, 6.0.4 with the IIci and Portable, 6.0.5 with the IIfx, and 6.0.6 with the LC and IIsi; the Classic can run earlier versions). While this worked to accommodate new hardware, it became somewhat of a pain for developers since many of them would design their software for a certain system version. For example, a program may only run on 6.0.4 or higher since it was developed with that system version. For some applications there are obvious reasons for a particular version of System 6 (6.0.5 for 32-bit QuickDraw, 6.0.7 for TrueType) but for others it was a matter of what was on hand or to provide guaranteed compatibility with a particular machine.
Back in the day one could upgrade to a new version by visiting a users group meeting. This was before online updaters were available (they would not have made sense at all given the low popularity of online services and the speed of modems circa 1989-1990). Apple continued to let users groups do this until 2001, when the current policy was instituted (the same one that prohibits sites from hosting old system versions). The problem with going to the users group was that meetings were usually only held once a month, meaning one may have to wait several weeks before being able to use that $500 program they just bought.
Apple started to do the same thing with System 7 when 7.0.1 was released to accommodate several new Mac models. This changed in 1992 when Apple started to roll out a new Mac every other week (if you remember the Performa line you know how many would come out per year; the quicker-changing landscape of technology in the 1990s was another factor since computers would become obsolete more rapidly than they did in the 1980s). This is why System 7.1 was released. It gave Apple an operating system they could add enablers to so that the same system version could be used for a few years. All someone had to do was toss in an enabler program and the same System 7.1 that shipped with a Classic II (which does not require an enabler) would work fine when shipped with a Quadra 840AV one year later.
System 7.1 lasted for about a year and a half before it became slightly obsolete (7.1.2 was released with the Power Mac). System 7.5 was a new "reference release" (as many termed it) that incorporated all enablers from 7.1 into its code and started over again with enablers. In other words, the LC III and Color Classic (both of which required enablers with 7.1) no longer needed them with 7.5 although subsequent models would have.
Some double decimal upgrades continued to exist as bug fixes, key feature additions (such as the addition of PPC native code), or as special feature sets (such as 7.1.1, also known as System 7 Pro).
The reference release made software development much easier. Instead of picking a random version of System 6, developers could choose to make 7.1 the minimum requirement and not have to worry about everyone having to upgrade. However, many did have to upgrade and had to pay for it--7.1 was the first version that users groups could not freely give away (keep in mind this decision was made in 1992 when Mac sales began to slip).
Interestingly, Apple has gone back to double decimal point upgrades for OS X, although with online updates the process isn't an issue (unless you're using dial-up or don't have web connectivity).
Hopefully this clears up some confusion about system enablers!!!