Usually when talking about accuracy the classical Fourier Uncertainty principle
enters the conversation (I would skip the quantum mechanics stuff). Essentially you can have good resolution in frequency or good resolution in time but there is a tradeoff between the two.
In order to go back to a time series from an image, you at a minimum need to retain phase information which is not included in the image. There is also the issue of how you map a magnitude level of a STFT bin to a pixel value, which usually involve some nonlinear transform (including clipping). The eye and ear are very different in how they perceive a level. It is much simpler to just keep a copy of the time series and use the image to index back into the section of signal you might be interested in.
The utility of STFT based images is that given the roughly 100ms delay that a human brain needs to perceive something, for audio signals, you can craft a real time implementation where you can simultaneously listen and look at the signal. One can use tricks like averaging, maxing , along with decimation to get a useful "real time" waterfall time/frequency spectral analyzer. These sorts of processing are specific to particular application. You would use different techniques for something like looking for low SNR tones in SONAR than those for clean music. This makes going from image information back to time series more difficult ( i really want to say impossible but one needs to be careful with that word).
The STFT is linear. The STFT of 2 signals is the sum of the STFTs of each.
are also linear and can produce images and relatable to the STFT
For high SNR signals, people have also created time/frequency images using AR models by stitching together segment line by line.
These typically have good frequency accuracy of peaks. but not so good at troughs.
Before I go further, there is an example that might be useful about redundancy in time/frequency images. Led Zepplin's, Stairway to Heaven is 482 seconds long. At a nominal 44100 samples/second, the time series is 21,168,000 samples. If I choose a 1024 point FFT at 50% overlap, I will have a 41,344 by 512 array, which I can reasonably create a scrollable image.
If you want higher "time resolution" you increase the overlap, so 75% overlap produces 82,687 by 512 array which can be scrolled but it listening and seeing simultaneously becomes more problematic.
One can use some tricks to reduce the scan rate of wavelet grams but it is harder if you want to listen and see simultaneously. The same is true of AR grams.
As mentioned in the other answer by @mathewpollard there is Wigner Ville (WV) which is nonlinear. WV is a special case of the "Cohen Class" which also includes Choi Williams.
I had the privilege of attending a joint lecture by Leon Cohen and Al Nuttal a long time ago. I remember a quote from Nuttal, "Wigner Ville is like small child, when good it is very good, when bad it is very bad".
A problem with WV is that it is nonlinear, there are cross terms. The image of 2 simultaneous signals is not the sum of the individual images (neglecting pixel level mapping issues) . There are artifacts in the image that don't belong there. There are modifications to WV that mitigate the cross term problem. People do use WV. You need to understand it.
The other issue (which may not be your issue) is the redundancy of WV. In its simplest form, you update it every new sample. The pixels will fly by your display faster than you can perceive them in human real time.
The strength of STFT based grams for audio signals is the ability to listen and see simultaneously. It is much easier for an analyst to relate the spectral feature to the sound it makes and identifying the cause . Understanding the data is more important in many cases than accuracy.