To answer the main question, no - there is nothing in Luke 11 which implies that others drive out demons by "the finger of God". To understand Jesus' words here it's beneficial to have some context around exorcism in the ancient world, and we will explore this below.
There is an implication that the Pharisees accuse Jesus of having spiritual power over demons by a connection with Beelzebub, but don't make such accusations against their own people. Jesus' point is more to highlight the hypocrisy of their accusation than to make any claims about the methodology or origins of the other exorcists.
Exorcism in the ancient world
Exorcism was a common practice for various groups in the ancient world, and probably covered a range of cases between genuine demonic possession to various degrees of mental illness. The Exorcist would apply something considered to be 'holy' or have spiritual power in order to forcibly drive out the unholy presence, such as herbs, oils, running water, or even the written name of God. Josephus gave one firm example:
"I have seen a certain man of my own country, whose name was Eleazar,
releasing people that were demoniacal, in the presence of Vespasian
and his sons and his captains and the whole multitude of his soldiers.
The manner of the cure was this: He put a ring that had a root of one
of those sorts mentioned by Solomon to the nostrils of the demoniac,
after which he drew out the demon through his nostrils; and when the
man fell down, immediately he abjured him to return into him no more,
still making mention of Solomon, and reciting the incantations which
he composed. And when Eleazar would persuade and demonstrate to the
spectators that he had such a power, he set a little way off a cup or
basin full of water, and commanded the demon, as he went out of the
man, to overturn it, and thereby let the spectators know that he had
left the man; and when this was done the skill and wisdom of Solomon
were shown very manifestly." - Flavius Josephus, Antiquities of the
Jews, Book 8, section 42
Thus there were understood to be many mechanisms for gaining power over demons, without invoking God's direct action.
Exorcism among early Christians
Exorcism became an extremely common practice in the early church. The name of Jesus or sign of the cross were taken as symbols of power which could expel demons. Baptism was in itself understood as an exorcising experience, which had the power to free believers from demons in their past, and the catechism process was expected to release new believers from demonic possessions of the past, which in some cases were issues like addictions to sins such as lust and anger. Among the early church fathers, Justin Martyr and Origen in particular emphasised that Jesus' Kingdom was one which in a primary sense overcame the demonic powers of the world.
"Christians still have a remedy against demons: they can drive them
out by prayer and lessons from Holy Scripture, not only out of men's
souls, but - and mark this - out of animals as well." (Origen, C.
Celsus. vii. 67)
Origen held that certain names, sounds and syllables had the very power to invoke the persons they represented, and so that the names of Jesus and the God of Abraham in themselves had power independent of the person using them. (Exh. Ad Martyr, 46)
As we ourselves understand the gradual decline and tendency of the early church towards superstition and mysticism/paganism, and understand that it eventually became relatively 'normal' for Christians to seek exorcism and deliverance regardless of whether they were genuine disciples, we can understand the relevance and caution of Jesus' words. As the pages of history turn, we find the growth of many who drove or claimed to have driven out demons, but in themselves did not 'do' the works Jesus commanded in his Sermon on the Mount which you quote in your question.
The distinctive thing about Jesus was that he did not need to invoke any secondary name or article in order to drive out demons. He wasn't recorded using any special names, formulae or items to gain power over demons - he could command them himself. And so he could say that it was the very 'finger of God' driving them out - not just a man invoking the power of others greater than himself.