Try a hard reset: Remove the battery, memory card (if installed) and disconnect any usb cables. Press and hold the ON/OFF button for 5 seconds . Allow the camera to sit idle for 30 minutes. Press and hold the ON/OFF button for 5 seconds again. Reinstall the battery without the memory card and try to power on. If functioning OK, reinstall the memory card. If the camera does not operate normally after this, forget the local shop, they really may be dumbos. The service at the link below can fix this model. Please check the listing for estimates.
Since you are out of warranty, considering the age and present value of this camera, paying the cost of having it repaired, approximately $125, if sent away, really depends on how attached to the camera you are but may not be really cost effective. The standard "rule of thumb" is if the repair cost is greater than 50% of the value of the camera, it is not worth it. A good used Powershot SX260 can be purchased for about $55-$80 and I've seen a number of new ones online for as low as $135.
Please keep in mind that my diagnosis & solutions provided are directly dependent on the accuracy of your description of the problem. As with any "do it yourself" fixes, success is a "team effort", since I can't see or touch the camera, and relies on the customer's manual dexterity and ability to follow the instructions well. If the "do it yourself" approach doesn't work out for you and you need conventional repair, please let me know.
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