The assumption that thought is an actual thing and that thought has a source, arises only in the realm of imaginary separation. There is, by definition, no way to determining the imaginary source of an imaginary thing.
Whilst it is true that thought is ultimately an illusion, and trying to get rid of thought is also more of the same illusion, there is a logical flaw in this objection, as follows: thoughts may be imaginary, but that does not mean their source is imaginary. Fiction or imagination can have a real source. The teaching is to locate that Source (the Self) and abide there in Pure Being, which is devoid of thought, until the tendency to imagine duality (ie. thought) ceases.
Here are two quotes to illustrate the above points, one from Sri Shankara, and one from Sri Ramana:
‘The binding, and the getting rid of bondage, are both mirages. The belief that bondage of the Real, is, and the belief that it has ceased, are both mere things of thought‘
~Sri Shankara, Vivekachudamani verses 571 and 572
‘All doubts will cease only when the doubter and his source have been found. There is no use removing doubts. If we clear one doubt, another doubt will arise and there will be no end of doubts.‘
~Sri Ramana Maharshi, Day by Day with Bhagavan