The recently released Player's Handbook Errata clarifies this:

Ready (p. 193). You have until the start of your next turn to use a readied action.

You can ready a concentration spell

When you ready a spell you completely cast it and hold the energy but none of the effects of the spell happen until it is released.

If your concentration is broken, the spell dissipates without taking effect.

The spell-specific concentration aspect doesn't take effect until its release just like the duration and other effects/attributes don't start taking effect or counting before it is released. The spell-specific concentration starts when the duration of a spell starts. And the duration of a spell starts when the spell takes effect (so upon release in this case).¹

Thus, since there is only one source of concentration active when holding a spell, there is no conflict and the Ready action proceeds as normal.

Also, regardless of the above logic, if Ready was meant to exclude concentration spells it would have come out and said so instead of relying of really minute logical inconsistencies to do so.

The Crawford ruling doesn't change anything here

Regarding the Crawford Tweet looking at the full sentence reveals its context and why it does not affect this case:

A concentration spell ends the instant you start casting another concentration spell, and a spell takes effect when its casting ends. If you cast invisibility in succession, you're visible during the 1-action casting time—effectively for a split second or so.

The ruling doesn't matter here because there is only one spell and one source of concentration happening at a time when you ready a concentration spell.

^{1 - If duration was counting during the held part of readying then it would be impossible to Ready instantaneous spells because their duration would have expired the instant after you started holding their energy. And the book explicitly gives the example of readying magic missile as a valid ready action. Thanks @Gandalfmeansme for this suggestion.}