The only currently available MILC I would even suggest to anyone as appropriate for shooting sports is the Sony α9, and only then when using the most expensive Sony lenses that allow focus tracking of moving subjects without reducing the frame rate significantly. Using cheaper Sony lenses or third party lenses with adapters slows even the α9 from 20 fps to anywhere from 15 fps down to as slow as 5 fps when focus tracking between each frame is required.
Other than the FE 100-400mm F4.5-5.6 GM OSS, none of the currently available f/4 telephoto lenses longer than 135mm from Sony allow 20 fps with continuous AF between each frame. You must use one of the top f/2.8 or faster Sony lenses to get AF-C and 20 fps at focal lengths longer than 135mm. The best the α9 will do with the Sony f/4 70-200mm and 70-300mm telephoto lenses is 15 fps, which is in the same neighborhood as the Canon 1D X Mark II at 14 fps with AI Servo AF. There are lists of lenses at the online Sony α9 support site that show what frame rates are possible with what lenses when AF-C is desired. The Sony cameras aren't as well suited as the top level DSLR bodies for shooting sports from an ergonomic standpoint, either.
Will mirrorless cameras be ready for "prime time" in the sports shooting segment in the relatively near future? They almost certainly will. But they're not there yet unless you have a boatload of money to spend on an α9 and some high end glass, and even then there are advantages and disadvantages when comparing the Sony system to similar DSLR based systems from Canon and Nikon.
There are reasons why there are still a LOT more top level sports shooters with 1D X Mark II and Nikon D5 systems than with Sony α9 systems. Many of those shooters could and would switch in a heartbeat if they thought it would give them a competitive advantage on the sidelines. The only place I see the Sony α9 making much headway is in the golf and tennis sectors, where silent shooting is a significantly greater concern than with most gym or field sports.
For the vast majority of MILCs, to get the fastest frame rates all of the frames must use the initial focus distance determined before the first frame is captured. That's almost totally useless for shooting sports.
It doesn't matter how many frames per second you can rattle off if they're all out of focus.
It doesn't matter how little noise is noticeable in the image if they're all out of focus.
It doesn't matter how fast and accurate the focus for the first frame was if each successive frame is further and further out of focus as your subject move away from where they were when the camera set focus for all of the frames in a high speed burst.
The highest priority of a sports camera will always be about speed, accuracy, and consistency of focusing on moving subjects. Always. That requires the ability to track moving subjects and refocus the lens between every frame! When that requirement is included in the mix, pretty much all of the MILCs other than the Sony α9 wilt in comparison to the best "sports" DSLRs such as the Nikon D5 and D500 and the Canon 1D X Mark II and 7D Mark II.
If you are having issues with a Canon Rebel T5i when shooting sports with an f/4 lens, your most practical solution is very likely not to go mirrorless and only improve either your sensor size or your lens' maximum aperture. The most practical thing would be to get an f/2.8 lens that works better with the T5i than your f/4 lens does or get a Canon APS-C or FF body designed to excel at shooting sports that works better with your f/4 lens. It would also be much more economical to find a DSLR "sports" body and faster lens that does much better with sports than a Rebel T5i + f/4 lens than to move to a Sony α9 and the lenses it needs to give what you seem to think you will get with less capable MILCs.¹
Full disclosure: I often shoot sports with a Canon EOS 7D Mark II and EF 70-200mm f/2.8 L IS II. Do I get the absolute best image quality possible with any camera/lens available by using such a combination? No, I do not. A full frame Canon EOS 1D X Mark II with an EF 300mm f/2.8 lens would give me better AF performance, higher frame rates, and better image quality at the expense of the flexibility of being able to zoom out. It would also cost me about three to four times more than what I have tied up in the 7D Mark II and EF 70-200mm f/2.8 L IS II. When posted online at typical web sizes, though, my work with what I use stacks up very well against what others are posting from the same sporting events using FF bodies with more expensive lenses.
If the light in a venue is extremely dim, I will move the 70-200mm f/2.8 lens, or even an EF 135mm f/2 L, over to my full frame EOS 5D Mark III and give up frame rate and some of the reach the APS-C format allows in exchange for better low light/high ISO performance. But one must keep in mind that if one crops an image taken with a FF camera to the same field of view that would have been provided by an APS-C camera with the same sensor technology, you're giving that low light/high ISO/low noise advantage right back.
Is the F2.8/APS-C configuration faster/more accurate than the F4/full-frame configuration?
That all depends on the specifics of the particular camera body and particular lens in question, and how each combination of camera+lens works together. There is no universal generality in this respect. Some FF + f/4 lens combos will outperform other APS-C + f/2.8 combos and vice versa.
What is more or less generally the case:
- With the same lens, a FF camera has a wider baseline for the AF system to leverage. If that wider baseline is properly exploited by the camera's designers, it allows faster, more accurate, and more consistent AF than an APS-C camera with a narrower baseline due to the narrower image format.
- With the same camera, an f/2.8 lens will outperform an f/4 lens in terms of AF speed, accuracy, and consistency if the camera's AF system is designed with focus points that exploit the wider baseline provided by an f/2.8 lens compared to an f/4 lens.
- To get the best performance one should use both a larger sensor and a faster lens. A FF camera and f/2.8 lens should always outperform an APS-C camera and an f/4 lens when they all use the same generation of technology for the image sensor and AF systems.
If you can only afford an f/2.8 lens or a FF camera, but not both, then you either need to consider waiting until you can afford both or you need to upgrade one before the other in a system that works with your existing other pieces.
Please do not misunderstand this as a flippant potshot or take it personally, but the truth is that knowledge, experience, and skill of the photographer makes a larger difference when shooting in difficult scenarios such as sports under low light than differences in gear do. For more on this aspect, please see the following questions here:
How do I diagnose the source of focus problem in a camera?
the best way to improve image sharpness on Canon 700D
Why are my football action shots blurry?
Recommended shutter speed for action sports?
Lots of noise in my hockey pictures. What am I doing wrong?
What Canon lens would be good for shooting high school/college sports?
When should I upgrade my camera body?
How to know you've outgrown your equipment?
Many of the issues with getting sharp images shooting sports under artificial lighting are similar to shooting concerts/theatrical productions. You might also find these questions helpful:
I'm having trouble getting sharp pictures while shooting a concert from a press pass location
Best ways of photographing at a concert/festival
Pictures of dancers on stage
With more sophisticated and complex AF systems also comes a steeper learning curve needed to master such systems and understand how they work and how the plethora of options available to the photographer impacts the results:
Why isn't my Canon 70D autofocus accurate in manual zone AF mode with a 50mm f/1.8 lens?
Pictures of surfers start in focus then go out of focus?
¹ Expecting every mirrorless camera to give the same performance advantages as the current MILC top-of-the-heap Sony α9 gives is about like expecting your EOS Rebel T5i to give the same performance as a "flagship" DSLR such as the Canon EOS 1D X Mark II or Nikon D5.