Yes.
You cannot concentrate on two spells at once, thus casting the second spell, ends the first. Because casting a spell that requires concentration ends the first, and the spell requires concentration for the duration of the casting effort, it's a spell that requires concentration.
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.
Best Answer
Twinned Spell doesn’t actually make you “cast two spells”, it just says that:
Note that the text says that it is the same spell. You can't concentrate on two spells as you say, the first spell's concentration would end because you would need to concentrate on the second spell.
Instead, treat a twinned spell as if casting a single-target spell like Hold Person that has the option to cast at a higher level to target an additional creature.
As for the concentration breaking: Since you are only concentrating on one spell, only one check is needed to fail for the spell to end on both targets.
For example,
Witch Bolt has the following mechanic:
It's your turn, you cast Witch Bolt and spend sorcery points to use Twinned Spell. You cast Witch Bolt only once but you get to target two creatures.
After making two attack rolls, let's assume you hit both creatures (because if only one hits then you simply resolve the spell like normal). Now, both creatures take 1d12 lightning damage and on your subsequent turns, you can use your action to automatically deal another 1d12 lightning damage to both creatures again.
The spell ends on ONE creature if the creature moves out of range or has total cover from you.
The spell ends on BOTH creatures if you use your action to do anything else except dealing the 1d12 automatic lightning damage OR you are hit and lose concentration.