1) Spend a bit of time and energy to assess the damage first, maybe charging a small fee. Essentially, as I understand, the owner had a choice of re-thatching the roof or opting for tiles instead as a substitute. The decision would depend on the economic expediency and the information provided as part of the initial assessment would have had value to the home owner in its own right.
2) Once the contract is signed it's not a matter of choice for the thatcher whether to continue with repairs on own expense, it a legal matter that can be in-forced by the cottage owner through court.
3) The thatcher could have bought professional indemnity insurance to delegate the risk to a third party.
4) The thatcher could have signed the contract on time and materials basis, explicitly leaving the risk of additional work with the cottage owner. In this case it would only be right for the thatcher to let someone else assess the damage for the customer to avoid the conflict of interests.
5) Finally the thatcher could have done the work on project basis charging more for the risk he takes and then hope that over a series of engagements he will still be able to make profit. As a variation he could have specified a contingency within his initial estimate agreeing with the customer that the contingency can only be exercised based on customer discretion should additional work becomes apparent upon stripping.
The problem here is lack of generic project management skills and experience on the thatcher's side and specifically estimation, contract and risk management.