If you do a search of this forum, and 2 other popular foreign teacher forums, you will find many reports from newly arrived (in China) foreign teachers regarding medical reports obtained overseas.

The recurring theme is that the new teachers wasted their time and a couple of hundred dollars undergoing a thorough medical exam in their own country (including a chest x-ray, ECG, HIV and syphilis tests + more) only to find that they had to undergo a second, thorough medical exam at a Chinese government medical center (paid for by their school) when they arrived in China.

It seems to me that the recruiter is very familiar with Chinese bureaucracy and was trying to do you a favor.

Anyway, the best way to learn something is to experience it oneself, rather than be told by another person.

If you do eventually secure a job in China and if you do undertake a proper medical exam in your own country prior to going to China, please DO submit another report on this forum as to whether you had to undertake a second exam after arrival.