" Do we ever stop learning?" 
Never, and in my case some things I need to learn is hopeless! You'll get there. You have to walk before you run so just keep shooting. In my case I need to crawl first.
jburch,
Google led me to some higher ISO images from your T5 and it still looks very acceptable at ISO 3200 and probably 6400 but try some different ISO settings and see how high you are comfortable going in terms of the ISO / noise tradeoff. You can definitely go above 800 with your camera and unless you are satisfied with images where your son is freezing in place prior to or after a pitch you are going to have to go with higher ISO at night or otherwise shutter speed will be far too slow to freeze motion. Even with a f2.8 lens, the exposure triangle isn't going to let you hit a point with low ISO and enough shutter speed in that venue.
If you aren't shooting in RAW, do so. That is a necessity to get the most out of your camera under difficult conditions and the same is true even if you move up to Canon's more expensive and capable bodies. If you aren't using RAW already, Canon has free software available for download (DPP) that will allow you to do post process work in RAW and then convert the files to jpg format for sharing.
Av (aperture priority mode) with fixed ISO was the best sports shooting mode available for many years and is what I always used with my 2005 era EOS 1D Mark II but that was because auto ISO wasn't available then which was a holdover from the film days.
Looks at the blogs and writings from sports shooting greats like Peter Read Miller and Scott Kelby and they have also embraced auto ISO as the mode of choice for shooting sports, especially under trying varying and night illumination conditions. The logic is simple, for low light sports you are going to use the widest aperture your lens offers the overwhelming majority of the time so it makes sense to fix that parameter. If you are shooting action, which is the majority of sports shots, then using a shutter speed too low for that sport just results in blurry images so you want to fix your shutter speed as fast as possible without forcing the ISO too high with the minimum shutter speed at least sufficient to freeze action for that PARTICULAR SPORT AND POSITION. You have now fixed two critical variables (aperture and shutter speed) of the three controllable variables in the exposure triangle.
So let ISO float so that the camera can adjust for a standard exposure by automatically adjusting ISO with respect to the best possible aperture and shutter speed values you have fixed. This is a very standard sports setup for current camera bodies with auto ISO capability which includes most of those produced in the last 10 years. It simply provides the best your camera will be able to do within the constraints of lens aperture and venue lighting.
For your next night game, set your aperture to the lowest number possible for your lens and try your shutter speed at 1/500 for a few shots with auto ISO and see what you get. If the ISO is at its upper limit and the image is still under-exposed, try dropping the shutter speed another step slower. If the ISO is reasonable at 1/500, see what it will do at 1/640. For example with night football, in the worst high school stadiums I drop shutter speed to 1/640, for decent I use 1/800, and for a great high school and many college I go with 1/1000. Lighting conditions always force compromises, your role is to choose the best possible compromise.
Before going to a new body, check to see how much better the T7 is at higher ISO compared to your T5 to see if it is a worthwhile upgrade. You may find that a different body is better. The primary ingredients for good night sports shots are an optimal location and a fast lens, without those two ingredients even a great sports camera body will be crippled in its ability to provide good images.
Sports shooting is already difficult because things happen so quickly and gets far worse at night due to lighting issues. And every venue has its own peculiarities. The regional softball playoff I shot yesterday is in a park that isn't well constructed for photographers. The outfield fence is very high and there is no high ground just beyond it but standing on my Pelican gear transit case gave me another foot of height which worked well. To get images of the pitcher and some of the infield action, I spent part of a few innings in the dugout entrance area shooting over the coaches heads while dodging the girls running out to retrieve fouls. It worked but it added complexities and distractions to shooting the game because my goal at sports events is to be as invisible as possible which doesn't happen when you are working in a busy area. The coaches and players were very gracious and wonderful to be around but my personal goal is never to be anywhere that has the potential for causing distraction.
Even in daytime, auto ISO is very useful. When I am at an event I will choose a shutter speed that will provide proper exposure at ISO 100 in the brightest part of the venue and let the camera adjust ISO as needed. I shot the third game in a regional softball series yesterday afternoon on a very bright and almost cloud free afternoon but shortly after the game started a thick cloud temporarily blotted out the sun and the camera increased ISO to 400 to compensate and this image of an inning ending throwout at first base were some of the very few at the game not in the ISO 100-160 range. Moments later the sun was coming back out dropping the ISO back to 200 and then 125 for a slide into second on the way to another run.
Capturing images in RAW format provides files which are pretty tolerant of over/under exposure and two stops of adjustment in post would have worked pretty well but I prefer getting things as close to perfect as possible in original and auto ISO is a big help in that department.
Rodger
