You are right in that you have an indexing valve and that you may need to get a pump start relay.
What looks like a valve on the pump’s output is likely a pressure regulator.
Good news is that recent software changes will allow you to setup a similar schedule with a Gen 3 controller as you use with your old controller. Alas you will not be able to use the Flex Daily schedule (the best one, in my opinion) to the fullest, until you upgrade your index valve to individual valves for each of your zones.
Another good news is that your old controller seems to include a 24VAC pump relay somewhere inside, not sure how easy it will be to get to it, but it may be possible to setup Rachio to drive your old controller instead of the old analog clock you currently have (thus saving you a few bucks on a separate pump start relay).
Alas I can’t find a picture anywhere of how your old controller looks under the front cover, when you find the time, see if you can safely turn off the power to your old irrigation controller and take a front cover off and share a photo of how it looks on the inside. We will be able to advise you at that point if conversion or replacement makes more sense as far as difficulty involved.