Hi and welcome to the valley.
I'm not sure what you mean by "marginal performance", but adjusting the antenna might help if you are missing some channels you think you should be able to receive. However, it sounds like you really want a guide and recording capability, so I wouldn't waste my time until you decide you're not going to get a cable card and subscription for your Ceton.
While Netflix and Amazon are great for their original content, etc., you might look into Hulu to see if it offers the shows you watch, but that still won't be like having cable with a guide, etc.
Since I bought a TiVo back in 2014, I haven't kept up with Cox offerings. I don't know why the Surprise store couldn't show you Contour and just explain what you'd get with their Flex Starter subscription. If they don't have anything set up to demo, maybe try the store on Bell Road in Glendale near 59th Avenue. From what I read, my impression is that Cox offers a starter package that offers the locals, some shopping channels and some music channels, but not in HD. To get them in HD, you have to upgrade to the Contour Starter service that you mentioned and I could not find a price for it unless it's the same as the non-HD package $10/mo. I'm not sure where you're seeing "live" cable channels, but I suspect that just means the channels you are paying for.
To get anything though, you will need a tuner or cable card. I don't see any reason you can't get the cable card instead of the tuner and still get the bundle pricing. I suspect your fear in committing to a 2-yr contract is that you may not be happy with the limited channels in the starter package along with Netflix and Amazon. You can ask, but I'd be very surprised if they wouldn't let you upgrade your service during the 2 years, though you might not get exactly the same discounted pricing you would get doing it now.
I'm not sure what you were told that leads you to believe the chat reps are telling you what you want to hear, so maybe further examples of questions/answers would help.
The TiVo Bolt allows you to connect to cable or OTA, but it only has 4 tuners and limited recording capacity. And you still need a subscription to get the guide and recording capability no matter what TiVo you purchase. While the lifetime is expensive to start, it becomes free after 36 months (compared to the monthly subscription). I'll be the first to admit that I think it's really expensive for just the local channels and before I bought, I decided that there were just too many cable shows we liked to go with the Bolt and got a Roamio Pro and now almost everything we watch is recorded, we sometimes need a 7th and 8th tuner. I should check into Hulu myself, but I don't believe they offer everything we watch.
Anyway, I pay $162, but ~$75 of that is internet and then another ~$17.50 is taxes/fees, so I'm paying ~$70 for TV. When I look at the number of shows and hours we watch, that's cheap entertainment. When my latest contract is up, I'll see what they have to offer, just like I do with SiriusXM, and make my decision. I'd love for my bill to be $30 like it used to be 20+ years ago, but that's not going to happen and we would not be happy with less service, so we bite the bullet and don't worry about it. And we're retired too, so we know what living on a retirement income is like.
Now, when it comes to TiVo boasts, it does interface cable, Netflix, Amazon, etc., to varying degrees of success. For example, Goliath has been available on Amazon Prime for months now and TiVo still can't find it. And every so often when they do an upgrade, all the episodes of some other programs show up again and I have to delete them. And if you watch an episode through the app instead of the My Shows list, it doesn't get flagged as having been read, so it has its inconvenient problems. Still, it's a great feature, as are Skip and Quick modes. It is nice to look up a show and find it's available in the guide, On Demand, on Netflix, Amazon, etc. I often find shows where it asks if I want to record it or watch it now via On Demand or streaming. Now that I have a 4k Smart TV, I do tend to use the TV apps for Netflix and Amazon mostly because we binge watch 3-4 episodes and it's just easier with the TV, but I love the TiVo and have not regretted paying the price. The thing is though you have to commit until you recoup the cost. I recouped mine in just over 2 years, so that part if all free now.
As far as the Bolt vs the Roamio, the Bolt is smaller and faster, though I don't know how much difference the speed makes because that certainly doesn't affect the speed of the show.
Like I said, you need a subscription for all TiVos. The Bolt does OTA or cable, the Bolt Plus does only cable, the Roamio OTA does only OTA, and the Roamio Pro (I guess still available through the outlet store) only does cable. With the Bolt, if you make a mistake, you can cancel cable after your contract and not lose the TiVo investment. The Bolt Plus can get expensive because it gives you storage capacity options up to 3TB.
I don't know if any of this helps, but hopefully it gives you more info to think about to ask more questions.