Why is it said that without quantum mechanics we would not have modern computers?

The reason is very simple. Computers depend on electronics. Even the first diodes and triodes that the first bulky computers were made up of depending on the quantum mechanical nature of matter. The present ones with the chip technology are directly dependent on energy levels and bands of conduction etc in the electronics used. Semiconductivity is a quantum mechanical phenomenon.

Edit after the editing of the question

What I mean is, at which point is that our modern understanding of quantum mechanics led to a technological development so fundamental for today's computers that we could not have got it working another way?

The crucial point where quantum mechanical calculations became necessary was with the use of transistor technology, which has morphed to chip technology. It was with the invention of the transistor that control of quantum mechanical calculations was necessary for the leaps in progress we have made. For the vacuum tube computers, it was not necessary except for explaining the tubes existence. The chip designs have reached the point of even needing to foresee the Casimir effect (QM vacuum between charged plates).

Why is it not enough with Maxwell, Bohr, Lorentz, (Liénard)?

Maxwell is not enough because the classical theory cannot explain atoms molecules and solid state. Bohr is not enough because the primitive calculations could not be used in complicated lattices. Lorenz is irrelevant for solid state physics, the energies of the ions and electrons are low.

I find this quite an imprecise catch phrase. It's as correct as saying with out quantum mechanics there would be no atoms because electrons would have fall onto nuclei.

There would be computers but not like the modern ones. The first (electrical) ones didn't depend on quantum mechanical effects, they used vacuum tubes in place of transitors. Not to mention you can make mechanical computers running even on water (I mean for signal instead of electrical current). Not very efficient though.

What they probably meant is that quantum effects lie at the basis of semiconductivity and solid-state transistors which led to a true electronic revolution. They made computers available kind of like Ford made cars, both made production mass-scaled and cheap.

EDIT: When you add "modern". It's a very vague term to me. Modern as in non mechanical - VC electronical, using high-intergration chips (solid state transistors), or nowadays modern?

I am not sure whether inventors of transistor used QM models to explain their work, or the inventors of 1st micro-chip. Maybe they didn't have to, they just needed find good materials. Never-the-less, hot it cannot be explained with out using QM, but this knowledge is not needed for things to work or invent and develop them.

Also, I am sure that today QM theories are needed and used to develop better and smaller transistors. These theories are used to simulate and design most basic building blocks of the most advanced chips that are being produced nowadays.

Adding the word "modern" to the title of the question completely changes it. In modern computers you need semiconductors, and the whole theory of solid state physics (band structures, doping, etc) is based on a foundation of quantum mechanics - since electrons in semiconducting solids behave in a manner that is more wave-like than particle-like, with each electron occupying its own distinct state. Making a semiconductor work well requires in depth understanding of these things.