Based on what you have said, it seems unlikely that an Employment Tribunal would conclude that you have been dismissed - you have been put on Gardening Leave until you reach 65 years of age. However an employer cannot now force an employee to leave at the normal retirement age - 65 in your case (so you indicated) - the law changed last year.
It seems unlikely that the employer intends to continue with the employment contract, so at some time in the near future (perhaps when you reach 65) they will dismiss you and at that point the evidence suggests that the dismissal will be unfair.
The way some employers deal with these situations is via a Compromise Agreement - basically an agreement to exchange cash for an acceptance that no claim for unfair dismissal or breach of contract will be pursuaded and the employee won't talk to anyone about what happened - this is what Twix was asking you.
Paying employees for Gardening Leave in advance sounds dodgy to me - I'd be interested in others views on this. For example, tax and NI will have been deducted at the date of payment. This may not have any net impact on you regarding tax (because you will be able to claim back the current overpayment in tax at the end of the current financial year, once you have 'earned' no more salary between now and April). But NI is a different issue - and they will have saved themselves employer's NI as well as saving you employee's NI if they have lumped all the salary into a single payment. If they have deducted the tax and NI as if they had been paid you normally (weekly / monthly) until May, they HAVE to have broken HMRC rules, not least because this period runs into the next FY.
I appreciate that none of this directly answers your question. Do I have a right to a written reason? - here we go full circle because most of us believe you haven't actually legally been dismissed.
If this is causing you stress, I'd be inclined to take others advice and just enjoy the early retirement. If it had been me, I'd want to get back at them, and a letter along the lines 'thank you for the payment dated x. Please advise if I have been dismissed and if so, the reason for my dismissal'. That should get them thinking.
I think you've been badly treated by an organisation that doesn't understand too much about employment law.