You can add a channel to your account via the Roku website, but it will only download and install to Roku models that are capable of running it. An individual Roku will only list a channel app in its channel search if the channel will run on that model.
Over the years Roku has added additional features and capabilities to new models as they are released. The developers of the channel apps, Peloton in this case, have to decide which older models without these capabilities they will code their app to support. On this page, Peloton says they support Roku devices that are Generation 3 and up.
Roku model generation information can be found in this Wikipedia chart: https://en.wikipedia.org/wiki/Roku#Feature_comparison
Over the years, Roku has used "Roku 2" in the name of several devices. You can get your Roku's full model name and number via Roku menu path Settings > System > About.
If Peloton's information re Generation 3 and up being required is correct, for Roku 2s this would limit Peloton to Roku 2 models 2720 (released in 2013) and 4210 (released in 2015). Earlier Generation 2 Roku 2s (all have additional letters in the name, 2 XS, 2 HD, etc.) would not be supported.
[Edited to add...] Re your installation lag question: You can force an update check manually to be sure you have everything on your account that can be installed to your unit via Roku menu path: Settings > System > System update.
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