All a compression algorithm has to do to get similar advantages to varint is notice that 0-valued bytes are common and compress them. A simple Huffman coding will do that nicely.
Is varint actually "way, way faster" than Huffman? I don't think it's entirely clear -- it probably depends on the implementation. Protobuf generated code inlines copies of varint encoding all over the place, which is better than not inlining, but still not very nice to the instruction cache. Huffman coding would process the entire buffer in one go and can be a much tighter loop. A lot of work has gone into optimizing Huffman coding, including with things like SIMD. You can't really leverage SIMD for Varints in Protobuf because the input is not a homogenous array. Varint is known to be a very branchy format which is not so great for performance. Branchless Huffman is a thing.
You can probably do even better with an algorithm tailor-made to look for zeros. I took a crack at this with "packed" format in Cap'n Proto, which I implemented in a branchless way, but not with SIMD (I'm no good at assembly). It's been like 7 years since I did the benchmarks but IIRC capnp+packing turned out to be pretty similar to protobuf in both size and speed... but there are a lot of confounding factors there. Could be interesting to replace Varint with fixed-width encoding in Protobuf itself and then run packing on the output and see how that performs...
But it really doesn't seem obvious to me at all that Varint would be "much faster" than a matching compression algorithm. Do you have some data behind that or is it just a hunch?
Disclaimer: I'm not a compression expert. I am the author of Protobuf v2 though. My recollection is that the original designers of Protobuf were never very happy with varint encoding. It was a casual decision made early on that became impossible to change later.