Based on what you've detailed you are predominately right on this matter in terms of your proposed course of action, but by way of a possible explanation on landlord's behaviour...
I don't think the landlord sounds overly confrontational, and clearly doesn't want to engage and is aiming to remain reasonably civil, while at the same time attempting to stay firm ,and there could be a couple of explanations for this
1) You mentioned they have no experience as landlords, don't know the ins and outs, and also no handyman knowledge at all. So given this, from their perspective they are likely going to trust a professional company that has given them a quote of sorts - exhorbitant as it is once the callout service fee is included - rather than a now ex-tenant, despite your reasoned rebuttals
2)The second possible factor is more speculative , but there may be some conscious minor level of resentment that has recently set in based on you being able to lock in your rent for 3 years during which time presumably rents increased,thus representing lost income for them in their eyes. This realisation may have only recently crystallized as they are no doubt in the process of working out what they may be able to charge the next renters based on the current market
These everday injustice matters, relatively minor as they are, eat at you I get it 100% ,but it may be worth taking the hit on this given your thoughts on future rental referrals
Guess it's about picking your battles and perhaps contemplating say in a couple of years time what path do you think will best serve your long term interests. But at the same time wouldn't blame you for digging in your own heels