Sanskrit alphabets are considered as deities and are present in the chakras. Hence Sanskrit is recommended for use in worship and mantras. If you look at the iconography of Mother Kali you will see that she is garlanded by 50 human skulls that denote the Sanskrit alphabets or deities. However, use of Sanskrit is not mandatory. Unfortunately the Swami does not quote any scripture for his assertion.
Among the Vedic sacred formulas, the Gayatri has been prescribed for
repetition from ancient times. The Vedic people had great faith in the
efficacy of mantras. The Mimamsakas considered mantras to be the
embodiments of deities. In fact, they accepted no deities other than
the mantras at all. To them Brahman is nothing but sound (shabda), and
sound produces form. Thus, the name embodied in the mantra is more
real than the form of the deity. .... Forms are made up of nothing but
fine vibrations, and vibrations are produced by sound only, so these
ancient conceptions are tenable. From another point of view, a mantra
gains spiritual potency through many years of being associated with
holy men who have repeated it and attained realization through it. The
concept of the mantra is based on the psychological fact that much of
our thinking depends on auditory symbols. Repetition of the mantra
creates a chain of thought which induces the mind with the thought of
God, and this is the aim of all spiritual practices.
................. Every Vedic mantra has a rishi who first intuited
it, a particular metre in which it is composed, and a deity to whom it
is addressed. In Tantric mantras the most important element is the
vija, a sacred syllable considered to be charged with spiritual
potency, as also Shakti or power and a kilaka, a pillar (inner
syllables on which the mantra rests.) All the letters of the
alphabets are considered as different matrikas, minor deities around
the main deity. The mantra is charged with a special potency, so
much so that the Tantrikas believe that when offerings are made to the
deity with the appropriate mantras, the deity accepts them
immediately. ......... The Chandogya Upanishad I.1.10 says about the
efficacy of the repetition of Om: Both perform spiritual practices -
he who knows and he who does not know. But knowledge and ignorance are
different in their effects. Whatever is performed with knowledge,
faith and meditation, becomes spiritually effective.
Meditation and other spiritual disciplines by Swami Swahananda
There is an extensive discussion of Vaidika sabda in "The Garland of letters' by Sir John Woodroffe.
In the Yoga Bhasya that is Vyasa's commentary on the Patanjala-Darsana
(3.17) it is said: Tatra vagvarnesvarthavatiti. According to this,
each letter is intrinsically Sarvabhidhana-sakti-prachita, that is
possesing the power (Sakti) to denote and connote all objects
(Artha). For as the Visvasara-Tantra says:'What is not here is nowhere'.
A letter (Varna) is the whole cosmos in miniature. One Varna differs
from another in the relative latency of the universe of Artha involved
in it, the universe being the same in both. They come to denote
special Arthas by virtue of the order (Krama) of their collocation.
...... But the ultimate philosophical question is, "who, if any
one, first said that this sound meant that thing?" The Indian answer
is that the relation of the word and the object denoted are eternal
and not a conventional thing, that there is a system of Natural
names, whatever in fact that may be.'
The Garland of Letters chapter IX by Sir John Woodroffe
What the above quote means is that the Sanskrit words of a mantra for deity worship are not mere human invention. The implication of using a non-Sanskrit language is that the word that is eternally linked with a certain object is not being used. If that is not a concern then one can indeed use any language.
Let me use a concrete example. Take the word AUM=OM. OM stands for Brahman whether Saguna or Nirguna. Is OM a mere human convention? Did some people choose to use OM as the symbol of Brahman? The answer is no. OM is the eternal sound symbol of Brahman. Can we replace OM by say GY? Yes, we can indeed replace OM by GY from the intellectual point of view. All of us would associate GY with Brahman by convention. However, is GY a mantra? The answer is no according to mantra sastra. GY is a mere human convention and will never be a mantra.
Let me sum up the situation as I understand it. If one prays to God then one can use any language. Sanskrit is not needed. If one wants to meditate then any language can be used. One can chant in any language. Sanskrit is not required. Texts which are recognised by many as scripture can be in any language. Sanskrit is not necessary. Sanskrit is necessary if one wants to use any mantra. Just think of Tibetan Buddhists. They do not accept the concept of Brahman. Yet they use OM in the mantra 'Om Mani padme hum. The reason is that the Sanskrit matrikas are present in every human chakras and Sanskrit mantras take advantage of this fact.