I think you will find that many of the older and cheaper 70-300mm lenses from Tamron and Sigma will not be an improvement, if any, from an improvement over your Tamron 18-400mm, that this either optically or in terms of speed and AF performance. Some, especially the EF 75-300mm f / 4-5.6 III, could be worse. I always have a Sigma 70-300mm f / 4-5.6 APO DG from the mid-90s. It was adequate for prints in 4 x 6, 5 x 7, or even 8 x 10 made from 35 film mm, but not so much from the point of view of how we tend to view high resolution digital images. Not to mention that there are firmware issues when used with an EOS digital camera. It is very likely that you are experiencing firmware issues with third party lenses much older than your 80D.
The models EF EF 70-300mm f / 4-5.6 IS and EF 70-300mm f / 4-5.6 IS II show a clear improvement over the Tamron 18-400mm at 300mm, even with guns wide open at f / 5.6 compared to Tamron already at f / 6.3 wide open.
Canon's two IS 100-400mm f / 4.5-5.6 L IS lenses offer better optical performance than any other product in the same range as Sigma or Tamron. Here's how the two guns on the left and center compare to the Sigma 120-400mm f / 4.5-5.6 right at 400mm:
Note that the old Canon 100-400 is about as sharp in the center of the frame as the newer version, but the new lens retains this performance even on the edges and corners, while the old one only dies not. The way you tend to shoot and fit will determine the importance of the difference you will make.
Are there other options that I do not take into account?
Yes. There are some others to consider.
In the department of good deals, there is only the series EF-S 55-250 mm APS-C. The EF-S 55-250mm f / 4-5.6 IS STM withstands the most expensive (FF) 70-300mm f / 4-5.6 IS II. The former EF-S 55-250mm f / 4-5.6 IS II is a step below and more at the top of the class of Tamron and Sigma's cheaper 70-300mm zoom lenses.
Although less time consuming, in terms of image quality and (mostly) speed and autofocus performance, the 70-200mm third-party lenses, as well as Canon's, far exceed the goals you have listed. The 70-200mm f / 2.8 lenses are almost indispensable workhorses for sports shooters who need the f / 2.8 aperture when shooting under or at night. In daylight, a maximum aperture of f / 4 will usually be sufficient. The Canon EF 70-200mm f / 4L (the oldest non-IS version) can be surprisingly affordable.
If you need good optical performance and you can live with slower autofocus, you can buy the oldest Tamron 70-200mm f / 2.8 (Tamron SP AF 70-200mm f / 2, 8 Di LD (IF) Macro (model A001)). . For optimal optical performance, the new Tamron SP 70-200mm f / 2.8 Di VC USD (Model A009) and SP 70-200mm F / 2.8 Di VC USD G2 (Model A025) are in the same class than the Canon 70-200 / 2.8 L IS II / III. The current Sigma 70-200 / 2.8 goals seem to be just a little bit below them.
Most Canon 70-200mm L lenses take Canon 1.4X Extensions very well with little IQ loss, although the AF bit Slow down. A 1.4X extension transforms a 70-200mm f / 2.8 into a 112-320mm f / 4 and a 70-200mm f / 4 into a 112-320mm f / 5.6. The only ones I would consider as shooting sports with a 2X extension, which turns a 70-200mm f / 2.8 into a 140-400mm f / 5.6, would be the 70-200mmf / 2.8L EF IS II / III.