Each of these is a refusal this regime lets a counterparty attempt, paired with the answer to it. Reading them before you write is worth more than any amount of polish on the letter itself.
"Regulation 261 does not apply to us"
high likelihoodThe carrier says the European Regulation has no application to a flight departing outside the European Union, and treats that as the end of the claim.
What answers it
That is true and irrelevant. You are not claiming under the European Regulation. You are claiming under the Georgian implementation of Regulation 261/2004 under the EU–Georgia Common Aviation Area Agreement, which is national law binding on carriers operating from Georgia, and which sets the same amounts. Name the national instrument in the first line of your letter and the answer changes.
EU–Georgia Common Aviation Area Agreement, Annex III
A voucher instead of money
high likelihoodThe carrier offers a travel voucher, often nominally larger than the cash figure, and presents it as the settlement.
What answers it
Compensation is a money entitlement. A voucher requires your written agreement, and you can decline it without prejudicing anything. Vouchers expire, exclude peak dates, cannot be transferred and are worthless if the airline fails. The cash has none of those properties.
Regulation (EC) No 261/2004, Art. 7(1) — Art. 7(1)
"It was a technical fault, which is extraordinary"
high likelihoodAn aircraft defect is presented as an event outside the carrier's control.
What answers it
A technical problem arising in the ordinary course of operating an airline is inherent in that activity and is not extraordinary. The exceptions are narrow: a hidden manufacturing defect revealed by the manufacturer or a safety regulator, or damage from sabotage or terrorism. Ask what the defect was, when it was found, and what the carrier did about it — an airline that will not answer has not discharged a burden that sits on it.
EU–Georgia Common Aviation Area Agreement, Annex III