How to properly select a transformer in terms of thermal rise
You have to understand which bits of this scenario are in whose control, and so which bits you and the manufacturer specify. Some bits are easy to do with calculation, some may require actual measurement.

Why should a transformer be run cool? It's not a silly question, it needs a detailed answer. (a) The windings are insulated, and wound on some sort of former. If they get too hot, the insulation and former degrade, causing eventual failure. The transformer therefore has an absolute maximum winding temperature rating. (b) If mounted in a plastic box, the mountings must stay below the softening point of the plastic. (c) How hot do you feel comfortable with? If any part of the box 'feels hot', it may alarm a user. 

The manufacturer knows the limit for (a), you don't know what grade of insulation is on the wire (and there are different grades, depending on cost). If you ask him the precise question, he should be able to answer. It's an absolute temperature, not a temperature rise.

You can determine the limit for (b), you know what you've put it in.

You seem to have preconceptions about (c) which is more open to sentiment. While 60C is too hot to touch for more than 'a few seconds' with a bare hand, consider that domestic radiators are limited to 82C surface temperature to avoid causing harm with a 'momentary touch'. 100C is not necessarily too hot for electronics, it depends on their specification. We worked to an absolute external temperature limit of 75C, with a warning sticker if it was above 50C, but would allow internal and junction temperatures to approach specifications, sometimes 150C.

The manufacturer specifies temperature rise for his transformer at a certain load. Load is easy to quantify and communicate with the user, which is why that's chosen. He doesn't know what your local ambient will be.

The temperature rise consists of two components, iron losses, and copper losses. The iron losses depend (almost entirely) on input voltage, and are independent of load. The copper losses depend on the square of the output current. This means that two measurements are needed to predict how the temperature rise will vary with load, an off-load measurement, and a full-load measurement. Running at half load will result in only 25% of full-load copper heating, with unchanged iron heating. If the transformer is intended to be 'good' when run no-load, it will have often have been designed to have much lower iron losses than copper losses (note this is not optimum for best on-load efficiency).

It's possible to estimate the winding temperature of a transformer quite well. Measure the winding resistance at ambient. Run the transformer until the temperature has stabilised, then measure again. The tempco of copper is about 0.4% per degree C, which means a 10% increase in resistance for each 25C temperature rise.

Consider a SMPS solution, smaller and cooler than an old-fashioned iron transformer solution.