From the perspective of implementation of conversion checking, it really depends on the specific setting.
For vanilla intuitionistic type theories without more exotic features (like cubical TT, observational TT or fancy rewrite rules), untyped conversion is generally quite favorable. Here, the only standard feature that untyped conversion does not easily support, is the full $\eta$-rule for the unit type. However, it still supports a weaker $\eta$-rule, which says that $t = \mathsf{tt}$ for all $t : \mathsf{unit}$. This might look ugly from the theoretical point of view, but it still covers many practical use cases, and we can simplify the implementation a lot compared to full type tracking.
It's also possible to shift work from the conversion checker to the elaborator: we can track whether types are judgmentally proof-irrelevant during elaboration, and mark irrelevant terms as such. In this case, the conversion checker doesn't have to track types. AFAIK this is related to how SProp is implemented in Coq. From the perspective of conversion checking performance, this is clearly superior. However, it can complicate elaboration in annoying ways. If we have metavariables and/or postponed constraints, we might not learn about the proof-relevance of a given term until well after it is elaborated.
For deciding $\eta$ for $\Sigma$ and $\Pi$, which is the great majority of practical cases, it makes little sense to do type-directed $\eta$-expansion, because the syntax-directed version is already complete, and much more efficient. For example, Agda has typed conversion, but it still uses syntax-directed expansion for records because of performance concerns.
If we allow typed custom computation rules, like the custom $\eta$-like rules in Andromeda, then typed conversion is a must. If we move to cubical type theories, then I believe that typed conversion is not essential per se, but it can be convenient in the implementation.
If we add strict Prop, that can be viewed as a generalization of full $\eta$ for unit types, and once again we have the choice of distributing work between conversion checking and elaboration. I personally experimented with conversion checking which is universe-directed but not type-directed. This seems to work well for strict Prop: we track whether a type is in a Prop universe or in a proof-relevant Type universe, and we can shortcut conversion when we are comparing proof-irrelevant things.
t ≅ fun (x : A) => u
it will checkt x ≅ u
because the lhs must have a product type. At no point we have access to the actual type of those terms. $\endgroup$