Mouldy blinds

I know the blinds were fine two years ago when I lived in the house and I cleaned them regularly.

Take a look at your Check-In report and if their condition is stated as you say you should have no arguments. What you know and what is formally written down (or photographed) are two entirely different things.

If you have no suitable Check-In report then it is your word against the Tenant's and - yes - the Tenant could easily claim they have not been used.. but that leads to the conclusion they were like this at the outset of the tenancy too.

Unless the house has serious problems... things like mould, condensation, damp etc. aren't the responsibility of the Landlord, more likely what we deem "Tenant lifestyle"... like some Tenants dry clothes on radiators and don't allow a property to have enough ventilation... or they heat it too much, or too little... or have long, hot showers in a tiny room with nowhere for the steam to go.

But - for you - the Check-In report should state whatever the condition was when the tenancy commenced.

...so it’s impossible for me to prove that they weren’t stained then.

This, therefore, is your answer.

I think it now depends on whether you want to maintain good relations with your Tenant. If you don't care you might try to force the issue to your advantage, either by asking them to foot the bill of replacement or intending to deduct it from their Deposit, but they could easily call your bluff... and, the fact is, you have no evidence.

Or you could suggest removal.
Or you could suggest cleaning as best efforts - you do say this was the request they made.
Or you could attempt an assumptive close by saying - we both know they were fine when the tenancy commenced (!) but as a gesture of goodwill may I suggest we go halves?

There's a line of morality somewhere in all this that you might not want to cross.

Yes, at the moment you are adamant they were fine when under your custodianship... but everyone can be mistaken... and, nevertheless, the Tenant claims they've not been used. Whether they should proactively clean something that they don't ever use (until now) is another interesting debate. Some Landlords might easily say "yes", it's part of the property the Tenant has rented.

Unsightly, stained blinds do not stop the blinds being what they are - blinds. They still function as blinds. That could be pointed-out to assess reaction. What kind of property is it, had you marketed it as "high-end"? If so, Tenants will probably have expectations in-line with that.

I think a hard line here will sour a business relationship... but you might not be especially bothered about that.

And a Landlord should always have a war chest for unforeseen repairs / replacements. What if the boiler went - would you be able to afford that?