Archive and backup can be two separate things, or one and the same - it depends on your particular needs.
An archive is not necessarily off-site/off-machine storage - it's an organized record that can be anywhere, including the internal HDD. The contents of the archive will vary, depending on many factors. It's very possible that you'd want to make a backup of an archive, no matter where it resides.
Backup is, first and foremost, disaster recovery. We think in terms of the big disasters first - total loss of the contents of an HDD. A whole-disk/whole-machine backup makes it easier to recover from disaster - just restore the backup to the new HDD. That's far less labor-intensive than trying to reconstruct the contents of the machine piecemeal.
Partial "backups" (saving only the files you think are important to you) almost always results in what I'd call a "Big Yellow Taxi situation" (from the Joni Mitchell song... "Don't it always seem to go, that you don't know what you've got 'til it's gone!").
Time Machine, like other incremental backup solutions, can resemble an archive - the ability to retrieve a single item from a particular date and time. Its user interface is designed to make that kind of individual file recovery as easy as possible. However, I like to look upon that as an added benefit, rather than its primary reason for existence.
You'll see lots of talk about using Carbon Copy Cloner (CCC) instead of, or as a supplement to, Time Machine. If you use CCC to make periodic, whole-machine backups while leaving Time Machine to do the incremental backups, it seems like a pretty good strategy. Restoring a whole-machine CCC disk image is very fast and straightforward if/when you need to do it. Restoring from any incremental backup (and CCC can also be used for incremental backup) will be more time-consuming, as the restore software has to reconstruct a final result from multiple backup sets. An incremental restore may have a greater risk of error since, with more "moving pieces," there's greater likelihood that one or more pieces may be missing. On balance, I prefer the up-to-date coverage of an incremental backup to the spottier coverage of, say, a week-old (or older) full backup.
Any automated backup system can fail, for any number of reasons. I don't know that an automated, incremental backup in CCC is any more reliable than a Time Machine backup, but I have no statistical info to back that up. The benefit of automated backups, of any sort, is that they're far more reliable than, "I'll remember to run a manual backup every day/every week." There are practical limits to how many whole-machine backups you'd keep, considering their size. While you may never need to go back more than a few weeks in time, if you feel that going way back in time may be useful for you, Time Machine is a far more economical approach.
If large external HDDs were expensive, that'd be one thing, but they're not. The usual rule of thumb for Time Machine external drive capacity is three times the capacity of the drive being backed up. So, a 1 TB internal drive should be backed up by a 3 TB external (that can be less than $100 at today's prices). It may be a very long time before you exceed that 3x capacity. I've been using my primary Mac for about 3 years. There's 1.1 GB on the 3 TB internal drive, the 3-year-old Time Machine backup size is 1.49 TB. Your mileage, of course, may vary.
I can appreciate the concerns that Time Machine may end up saving dozens of copies of the same file over the course of time (presuming you give that much attention to a single file), but the vast majority of files you work on are not going to get that kind of attention, are they? If they're works-in-progress for clients, one hopes that you're making enough to cover the cost - the time you may save retrieving even one of those interim versions may be worth the cost of storing interim copies of all your files. Further, since Time Machine consolidates backups over time (discarding hourly backups in favor of a consolidated daily backup, discarding dailies in favor of a consolidated weekly backup, etc.), many of those saved, interim versions of files will be discarded eventually, automatically.