We use those networks of components in engineering because its actually rather hard to do the time delay circuit you describe. If your input is a sine wave, you can detect its frequency and determine the delay needed to provide an anti-phase output. However, if your input is more complex, like a sum of multiple sine waves, or even worse... real inputs, it can be very difficult to tease them apart and get the phase behaviors correct. Those networks surrounding an op-amp are known to have very specific phase characteristics, regardless of the frequency spectra.
You can also design those networks to the specific dynamics of your object. Sometimes the best damping isn't precisely a simple anti-phase relationship, especially if your materials have non-linear behaviors.
You can, of course, use a microcontroller for this. There are digital filters which are designed to emulate the analog filters in those electronic networks. However, they are expensive. In a business setting, you never use a 70 cent micro-controller for what you can do with a 20 cent op-amp. They are also much easier to manufacture and have much better longevity.
If you are doing a study, you may wish to have flexibility. It is much easier to modify an algorithm than it is to re-spin a circuit board. For study purposes, a microcontroller may be the correct tool for the job.
One thing to consider is the output. A microcontroller is not well suited for outputting the analog signals you need to stabilize the system. You need some sort of D2A (Digital to Analog) device. All of them cost money, which is why you don't see them in industry unless there's a real advantage. DDS based converters, in particular, are very effective for this kind of output, but are comparatively expensive.
Also, remember that most of these devices produce signals, but not power. They tend to be unable to drive loads. You need an amplifier attached to them to take care of the load. This often involves a similar network of resistors and capacitors to properly bias the amplifier, so you are likely to find those networks appear again, in the end.