Well, I know that you have to have skill behind the camera. That's a given.
Also, I'm sure I would learn very quickly how to control the flash. I don't plan on taking pictures on indoor rides really anyway, but if I did, you can bet my flash would be off. As a Disney World veteran, nothing annoys me more than flashes on POTC and Haunted Mansion.
My current point and shoot is about 8 years old. Nothing in particular is wrong with it, but I just feel like an upgrade is in order. This is an investment, so I want to feel like whatever camera I choose will last and help me produce nice pictures.
The SX260 is a very nice compact superzoom. In good light, it can take perfectly adequate pictures. It's fairly easy to use. (Though as said, with any camera, extra knowledge goes a long way to improving your pictures).
In regards to your flash issue --- It's a non-issue. If you're worried about handing it to someone, just hand it to them with the flash already raised. Though personally, I hate pictures taken with built-in flashes anyway.
The T3 would be more of an -investment-. And a first step investment, at that. If you want an all-in-one final product, the SX260 may be a better choice. If this is the first step of a continuing investment, the T3 is the better choice. As your needs change, as your experience grows, you supplement the basic T3 starter kit with things like extra lenses (A telephoto lens, to get more zoom, a prime lens to get low light photograph, maybe an Ultra wide angle some day, maybe a macro lens, maybe a faster zoom lens).
The SX260 -- right out of the box, will have a zoom equivalent of 500mm. You would likely never get a 500mm Canon lens.... and just getting a lens with a 250-300mm equivalent, would be an extra investment.
Though telephoto zoom is over-rated by many camera purchasers, it is a factor.
So with the T3, you'd find yourself adding lenses. You may find yourself adding a tripod. (Though even with the SX260, you may want a tripod, especially for fireworks).
With the T3, you may find yourself adding an external flash, to take more natural appearing flash pictures than you can achieve with the built-in.
While you may invest in photo editing software with any camera... With the T3, you'd be more likely to consider shooting in RAW format some day, making it even more likely you'd invest in photo editing software.
So if you're looking to just replace your old P&S, the SX260 may be the right choice for you. It's a pretty good little camera.
If you want the best possible image quality, and especially if you want this camera to grow as an investment, the T3 is a better choice.
But to put it in perspective... With a dSLR, the camera body sometimes ends up being the cheapest part of your investment. It's not uncommon to own a $400-$1,000 camera body, and $1,000-$5,000 in additional dSLR equipment.
I personally spent $400 on my current dSLR body, but I have over $1000 in lenses, tripods, flashes, etc.