That quote, and its context, is not an argument that animals do not have exceedingly rich inner lives, nor is it an argument that language is some prerequisite for such a state. It is an effort to address, and dismiss, the kind of "17th century delusions" that you're worried about (and, in fact, participating in, whether intentionally or not).
The statements, "Animals are sentient", and, "Animals are not sentient", are opposite positions extrapolated from the same axiomatic premises; that there is some thing called 'sentience', that humans 'have' this 'sentience', and animals either do or do not. Most commonly this kind of thinking also assumes that there is one kind of thing called sentience, (often referred to in religions as a soul). Your phrase, "lack certain tooling", is in line with this logic; it implies that there is some absolute definition, some core kernel of "inner experience" that is separable from ancillary layers that it may leverage. This is analogous to the statement that a computer may be running an operating system, whether it has a keyboard and speakers or not.
What Dennett argues, here and in general, is that this definition of sentience is not nearly holistic enough. That there is no one objective thing called 'sentience'. Instead, there are a bunch of different things; being a lion, being a bat, being a human. Each of these is a model of experience in and of itself, and is defined in and of itself; you can't factor out a common set of experiences or functions and expect them to translate. Part of being a human is being able to, amongst other things, see, hear, taste, and speak. These are not tools built on top of some common 'sentience interface'[1]. To continue (dangerous and leaky) software metaphors, it's much more like a spaghetti-coded monolith[2]; the features -are- the system, and their implementation is part of a feedback loop into the deepest parts of the system, and all the way back out again. The tools aren't 'used by' the 'sentience'; they _are_ the sentience, and the sentience is the tools. A practical demonstration to consider here is something like synesthesia, where what might superficially seem to be well-defined verticals are crossed and interwoven.
And so, when he (extending Wittgenstein) says, "If a lion could talk, we’d understand him just fine. He just wouldn’t help us understand anything about lions.”, he's saying that if such a thing as a talking lion existed, it wouldn't help us understand lions because talking lions and non-talking lions have different experiences, in part because they have different capabilities, different available tools, and thus different architectures. This isn't to say that they don't each have some rich inner life. Just that these inner lives are not mutually comprehensible, shared, or even compatible. In fact, the article, further down, pretty much says just this, albeit in many fewer words:
“If you think there’s a fixed meaning of the word ‘consciousness,’ and we’re searching for that, then you’re already making a mistake,”
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[1] This line of thinking is very close to the "17th century delusion" of something like a homunculus. It is made more palatable by using a word like 'soul', however
[2] or, indeed, something like a neural network; the weights 'mean' something inside the system, but they're all relatively defined, not absolutely; you can't necessarily look inside the system and expect to understand what individual components 'mean' without addressing their entirely relative context.