Argthtjtr

Given that the goal is to approximate the ideal filter with an IIR filter, why would it be easier to design it in the continuous time domain than the discrete?

This paper says:

Discrete
time IIR filter design can be a very complex procedure in discrete time domain. Therefore, transformations have been
developed to use well known design methods in continuous time domain.

This may be a very trivial question (sorry if it is), but I don't see why it is easier.

Perhaps I could be asking this in a better way:
Why do we need to translate filter specifications from discrete to continuous time and then do the calculations in continuous time? Why not directly in discrete time?

This sounds like a quick conclusion.

It is true that discrete-time IIR filters are typically designed by transforming continuous-time filters (also valid for actual and used tools, for example MATLAB).

The research made on continuous-time filters has brought many results and analytical tools to design them efficiently, while the bilinear transform quickly makes them discrete.

The combination of both makes for a quick and efficient solution, which is used as a base in filter design tools, that also implement a numerical optimization routine to minimize the filter order while keeping it in the specifications.

Finally, I'll agree with other answers here; the complexity of the procedure may be high, but the computing power available nowadays makes it irrelevant. Least Pth-Norm Optimal IIR Filters are completely designed in the discrete domain (no transformation involved), and have more degrees of freedom than the usual transformed filters.

The term “easy” can mean simply to need less effort but can include cases where an accumulation of prior efforts can be leveraged.

Analog filters have a long history and someone in 1970 who was trained in analog design might consider modifying what they know to build a digital filter as “easier" than using an optimization routine written in Fortran running on their IBM mainframe.

There are methods to directly design discrete domain filters, and they are more modern. However, due to the legacy, historically first developed continuous domain filters are still used. They are arguably simpler, but they have significant drawbacks. They are easier, since they do not require extreme knowledge of digital IIR filter design. however, MATLAB, Octave and other DSP software does have enough tools to effectively design discrete IIR filter. Please see the following as a nice introduction to both FIR and IIR design. http://ece390web.groups.et.byu.net/dokuwiki/lib/exe/fetch.php?media=filter_design_notes.pdf.
To summarize: designing in continuous domain first, was easier earlier in time, since the tools for IIR filter design have not been available to the extent that they are available today. At the time, the alternative was to get involved in writing your own IIR filter design software.