I guess I am struggling with the 2nd argument which was there is no limit on the number of modes of vibrations that can excited? Why would there be no limit? Can someone please explain?
This follows from properties of the standard Fourier series expansion of function on a real interval; in order for the expansion to be able to reproduce variations of the original function (here, electric field) on a very short length scale, very high frequency waves have to be included in the Fourier sum. If variations on arbitrarily short length scale are to be captured by the Fourier series, then the Fourier sum over frequencies has to go over arbitrarily high frequencies $kf_0$, $k=1,2,...\infty$.
So the series terms corresponding to arbitrarily high frequencies (and thus infinite number of frequencies and terms) follows from the assumption that the physical electric field can have features on arbitrarily small scale. This may not be true physically, and there may be a short length beyond which there is no such features, and then the number of required frequencies would be finite, and the UV catastrophe avoided. However, nobody has found evidence for existence of such length; electric field seems to be able to have any frequency, even arbitrarily high one.
However, there is a good physical reason to expect suppression of the Fourier expansion coefficients after some very high frequency in case of equilibrium radiation hold in a real cavity; such cavity can't hold radiation of very high frequencies inside, because no cavity can be perfectly reflective to radiation of arbitrarily high frequencies. Even polished metal cavity will partially pass through radiation of high enough frequency (X-rays, gamma radiation), so achieving equilibrium at this frequency in the cavity would be very difficult; the cavity transparency will work against that.