Doesn't work on the 70-300L, but thanks.
If you can find a used Kenko Pro 300 DG 1.4x (NOT DGX; I can't vouch for its current firmware), it is an optically good TC, and may work well with the 70-300L, but EF at f/8 with a 3rd-party TC will result in a slowing of large changes in focus distance especially in low light, but should jump into focus quickly enough for small changes in focus distance.
If you have a local store that sells used optics, they might have one that you can try before buying.
The only problem I've had with the Kenko on the R5 and R7 is that if it is stacked behind another TC, many lenses will have weird AF and IS motor behavior, or trigger an error. As the only TC; no problems so far.