Does use of the term "Gordian knot" imply a heavy-handed solution?

You are right about the origin of the phrase: it comes from the legend of the complicated knot in Gordium that was hard to untie, and also associated with the legend is the story of Alexander cutting through it; thus the phrase cut the Gordian knot.

However, the phrase Gordian knot can be (and is) used by itself without any reference to cutting it, or brute-force solutions. That is why you have the definition you found, as "An exceedingly complicated problem or deadlock". Wiktionary also defines it as

(by extension) Any intricate and complex problem.

which is what you want. Your original intuition that led you to pick the phrase is fine; the only thing you'll have to worry about is how the phrase will be received by your intended audience (which for that section is your supervisors, I guess!), and you know that audience better than any of us.

If you search the British National Corpus, you'll find:

CDY 2498 He could sit with his very big brandies when Claire had gone to bed and talk about the Gordian Knot of the Arbuthnot relationships.

CM8 264 This cuts the Gordian knot.

CMT 103 A GORDIAN KNOT?

CMT 182 If Gorbachev expects to untie the Gordian knot, he has over-estimated his powers.

EW6 931 A similar attempt to cut the Gordian knot of Chapter 4 also proves to be fallacious.

HLB 6 Both sides expressed satisfaction with the result, and UN Secretary-General Javier Pérez de Cuéllar stated that "the Gordian knot has been untied" .

JY7 1476 They had joined a Gordian knot of vans, taxis, and automobiles that was inching forward at a pace that had set that little muscle in his jaw to knotting and unknotting.

K8X 1008 In recent years governments, particularly the Conservatives in Britain, have tried to cut the Gordian knot by imposing cuts in state support and forcing the railways to adapt.

So you see that a fair fraction use it just to mean something complicated. For instance "Gordian knot of vans, taxis, and automobiles" just refers to the usual gridlock, and "cutting" this particular knot would have no natural meaning.

Looking at the Corpus of Contemporary American English shows a lot more results, the majority of which don't make any reference to cutting:

that will drive a strong economy. A strategic plan should: # Address the Gordian knot of overly complex and contradictory fiscal measures that evolved from constitutional, legislative and

's first impression was that this intricate framework resembled a gigandc metal version of the Gordian knot. But closer examination

the National Institute of Standards and Technology the time it needs to untangle an emerging Gordian knot of overlapping technologies. (The ARRA provides funding for this effort.)

. If Sweden's example is any indication, the imprisoned Uighurs present a foreign-policy Gordian knot. # The men are members of a largely Muslim minority in western China

dispute survived him. Bill Clinton devoted much of his tenure to picking at this Gordian knot. He failed. George W. Bush failed at everything he tried. This

Daily, a professor of mechanical engineering. '' But if anybody can untie the Gordian knot of funding in this state, it would be great. If he does

a Chinese finger trap, a jigsaw puzzle, and a reasonable approximation to the Gordian knot. Near the bottom, lest she miss the point, was a CD-ROM

Like virtually all the other problems faced by the airlines, that remains a Gordian knot. # '' We need something going for us as an industry, ''

his congressional counterparts wouldn't be wrestling, as they are now, with a Gordian knot of methodological problems. But when the Bush legislation was first introduced in January

over the next few years. Yosemite National Park officials, though delayed by a Gordian knot of planning and approval processes, hope to have a transit system in place

over the next few years. Yosemite National Park officials, though delayed by a Gordian knot of planning and approval processes, hope to have a transit system in place

for the Salinas administration will be legalizing ejido land rights. This means untangling a Gordian knot of claims. Of the 28,000 ejidos, only 3,000 communities have legal titles

very least, would have yielded more mature theological data with which to untie the Gordian knot of theological questions. Part of the argument of this essay has been that

etc., etc.

There are also the citations in Wiktionary:

1785, Thomas Jefferson, Letter to Peter Carr (August 19, 1785): The knot which you thought a Gordian one, will untie itself before you.

1825, William Hazlitt, The Spirit of the Age, The Late Mr. Horne Tooke: Mr. Tooke thought he had answered this question satisfactorily, and loosened the Gordian knot of grammarians.

Your original quote is grammatically fine, and means what you intended it to mean.

Regarding your title question and sample sentence: Although using the term Gordian knot does not, as such, imply that your advisors recommend solutions that are heavy-handed or brute-force, it remains open to that interpretation. (Note, another question to ask is whether your advisors would regard "a better team of consultants is hard to imagine" as a compliment.)

I suggest you avoid ambiguity and circumlocution like "is hard to imagine" and "when it comes to". Instead, directly say what you mean to say; e.g., "My advisors and consultants have provided decisive help and critical insights in solving theoretical and technical conundrums that arose during this work." Posting a question on writers.stackexchange can give you further suggestions for alternative wordings.

I do not connect a brute-force solution with what you wrote. A brute-force solution is one which mindlessly tries every possibility until the solution is found. Nowhere did you imply that kind of mindless approach.

‘Brewer’s Dictionary of Phrase and Fable’ confirms the story with a little more detail, but Alexander, rather than untying the knot, cut it with his sword. Thus the meaning of to cut the Gordian knot is, in Brewer’s words, ‘to get out of a difficult position by one decisive step; to resolve a situation by force or by evasive action.’ In the light of that, you may want to reconsider whether it is appropriate in your thesis. You might also want to examine whether you need to make a classical reference at all.