It is worth recalling that the Litany of the Saints existed at the Vigil in the preconciliar rite, both before the 1955 revision and after. Before 1955, there was never any baptism at the Vigil, and yet the water was blessed and the Litany was sung. It was associated in people's minds with the blessing of water, even if that water would never be used for baptising anyone but for the replenishment of stoups, etc. The memory of baptisms at the Easter Vigil had been dead for many centuries.
From 1955, provision was made for a baptism (presumably of an infant) to take place, but I never heard of anyone making use of this option. The Litany was used as a "filler", half of it covering the changing of vestments and the lighting of candles, unveiling of statues, etc. It was not until the RCIA was promulgated in 1972, available in English from 1974, that baptisms generally took place at the Vigil.
So I tend to disagree with John Ainslie when he says that the link with baptism is what determines whether the Litany is sung or not. For many centuries it seems to have been linked with the blessing of water, for whatever purpose, as well as covering scene-changing activities.
Because the font in many churches was (and often still is) sited in a difficult place for liturgy, temporary vessels were routinely used. I vividly remember the huge cauldron used in the parish where I spent most of my childhood. After the water was blessed, the cauldron was hefted onto the shoulders of a number of burly sodality members, vested in cassocks and cottas, who then carried it down the church to the baptistery at the back and poured the water from it into the font. The huge slooshing sound was guaranteed to rivet even the most sleepy congregation.
I therefore also disagree agree with Peter Jones, who I think is saying that it's the font which determines whether the Litany is sung or not.
For me, it is much simpler:
Are there going to be baptisms? Then the Litany is sung.
If no baptisms, is water going to be blessed, whether in a temporary vessel or the church font? Then the Litany is sung.
If no baptisms, and no font, whether permanent or temporary, then no Litany.
Is the place where the water is to be blessed on the sanctuary? Then the introduction to the baptismal liturgy comes first and the Litany follows.
Is the place where the water is to be blessed somewhere else? Then the Litany is sung in procession to that place, followed by the introduction to the baptismal liturgy.
A further thought:
This is of course not the first time that the Easter Vigil rubrics have been found to be confusing. In the previous (1969/70) Holy Week rites, the Paschal Candle was lit from the new fire immediately after it had been blessed (p. 187, para 9 of the previous Missal). Then the incisions took place and the insertion of grains of incense. But then, on p. 189, para 12, the priest is instructed to light the candle from the new fire again!
This was in fact the result of editorial sloppiness, but many priests followed the rubric, lit the candle on page 187 and then got spattered with wax while they were incising and inserting.
The fact is that the incisions and insertions were all optional (they are printed between lines, accompanied by clear statements that they might be done, or that episcopal conferences were free to adapt or substitute other rites). When they were completed, the candle was lit. All very practical, and avoiding spattering, etc, as already mentioned..
Clearly, the rubric at the foot of page 187 does not belong there at all, but should have come after the optional rites — i.e. at the head of page 190, before the procession of the candle, where it will be read when the options have been omitted. In other words, the whole thing was an editorial layout error. The 2010 Missal is notorious for layout errors of this kind. The revisers have evidently no experience of publishing work.
Coming back to the 2010 Missal, it seems clear that, while the insertions remain optional (para 12: "the Priest may insert..."), it appears that the incisions are no longer optional (para 11: "the Priest...cuts a cross" etc). The language has changed, and the telling phrase in the previous edition "This may be done as follows" is no longer present. Could it be that the Congregation had heard about priests omitting the incisions because cross and symbols and year were already on the candle in the form of a Hayes and Finch decal? We will never know.