This isn’t impossible but due to the way each federated platform is structured it does make it more difficult to fold into a single app. What you’re looking for, perhaps inadvertently, is essentially a fediverse browser. In that case you’re almost better to use a regular web browser with some plugins that may better enable navigation of each platform, as this ensures you’re keeping it to one app.

I say almost because your second point makes that less viable, but in itself this would prove a thorny issue regardless as each platform handles filtering/blocking differently, as you’ve found. I think the latest version of Lemmy has tried to improve on this, but not every instance has updated due to a mixture of details in how different admins/teams want to approach it.

Also, unfortunately iOS further impedes this, given that many of those interested in the fediverse space tend to lean towards Android, so many of the more flexible apps, e.g. Fedilab are Android only. If there are iOS alternatives, I’m simply not aware of them, nor do they surface on a light search.

Unless someone else knows of something for iOS like what you’re after, I think for now the “best” approach for navigating different fediverse platforms in a single app may be via web browser and hoping they have a decent mobile UX. However for the best UX you’re probably going to want to use multiple apps for each federated platform and toss’em in a fediverse folder/tag on your phone.