That is not how some are reading it. It's not how I've understood Article 28(2) either, though getting new consent each time is nuts (e.g. I change from Mailgun to Postmark and have to ask all SaaS customers for consent).
Some discussions: https://seqlegal.com/blog/article-28-gdpr-problems-processor... https://seqlegal.com/blog/gdpr-sub-processors-and-authorisat...
If a company legally holding private data (i.e. a controller) gets a different subcontractor (a processor), they do not need new consent from the user; and the controller is still fully responsible for the data privacy.
In your example, you're a processor who operates the data with controller's authorization (let's avoid the word "consent"; this is a quite specific term in context of GDPR with a different meaning) - and they are responsible for ensuring that you will safeguard this data properly, so they can't transfer data to you if they don't have solid guarantees about what you'll do and not do with the data they're responsible for.
And the particular situation seems reasonable to me. You do not need to ask all SaaS customers for some action; as your links directly quote ".. or general written authorisation of the controller" i.e. your agreement can specify (if your customers are okay with that) that you're allowed to use other processors and change them. You do have to inform them about such changes (just as you informed them about the current subcontractors touching their data before they signed up, right?). If you change from Mailgun to Postmark, all you have to do is to send a notification to all your SaaS customers, and nobody is going to cancel over that. On the other hand, if you change from Mailgun to NigerianPrinceMailings Inc., then they might reasonably want to decline, so that's why the notification is required.
This makes sense from the user's perspective. Maybe they trust Mailgun but do not trust Postmark. If they have explicitly agreed to you sharing their personal data with one company you shouldn't be able to start sending that customer's personal data to another company without their consent.
If you sign up for my service and I ask for consent to send specific data to SecuriCo for "user analytics and tracking" I shouldn't be able to change that to sending the data to the NSA without telling you. The whole point is that the user should be in control of what businesses are doing with their personal information.
If you have the same use case of private data but have technical changes (i.e. a different subcontractor handling the processing) then you do not need a new consent.
If you have a new use case, then yes, you get to use only the data of those users who agree to it. Which will not be all users anyway, as consent must be freely given, i.e. with an opportunity to refuse consent but keep the service.
The expected result of "a consent box a lengthy disclaimer will ask consent for every piece of information and blanket every length of scope" would be the TL;DR reaction - since all of that must be opt-in, the user would just click "Continue", keeping the default settings that don't give you any consent.