Without a doubt, there are definitely some taxonomists who have memorised all of the species within their taxon of expertise (I’ve been in the field with several, and it’s definitely a great learning experience). However, these cases inevitably involve specialisation in a higher level taxon with a relatively low number of species. Unless you specifically only work on a very small clade, there are just too many species to remember, so in almost all cases, taxonomists don’t memorise every species within their sphere of interest, as it’s just not feasible.
Some taxonomists (often botanists) will work across multiple, sometimes very disparate taxa, and so all of the species within their remit might be very different to each other. Even if working within a single genus, memorising all species can be a daunting task; there are many genera (mostly invertebrates or plants) with hundreds or even thousands of species in them, and families with tens of thousands of species!
If I found an obscure weevil and showed it to a weevil taxonomist, they may well be able to ID it off the top of their head if it’s a distinct species/one they happen to recognise. What is more likely to happen, however, is that they are able to recognise the weevil to a coarse level like subfamily or tribe, and then use an appropriate key to identify it. And the reason they can do that (in addition to the fact of having access to/knowing how to access identification tools/resources like that in the first place, which is a whole other discussion) is because they have two skills they have developed over time:
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They can identify/sort specimens to coarse taxonomic levels by sight. This is a useful thing you can practise now without being an expert: understanding the differences between coarse taxa such as order or family. If I see a photo of an insect with only two wings, with the other pair reduced to halteres, it doesn’t matter what species it is or where it’s from, I can tell you that it’s either a fly, Diptera (if the wings are the forewings), or a twisted-wing parasite, Strepsiptera (if the wings are the hindwings), and eliminate all other insect orders, because I know this important character. If I see that the insect has piercing/sucking mouthparts, I can straight away eliminate groups like Coleoptera and Orthoptera because they have chewing mouthparts, and narrow down the options to groups such as flies or bugs among a few. You might not realise it, but you, and pretty much everyone on the planet, already has this skill at some level! If you were to show a photo of a dandelion and a photo of a tiger to someone, they could tell you that one was a plant and one was an animal, i.e., they’ve just identified/differentiated them at the level of kingdom. From that basic starting point, you can start working your way down the taxonomic ranks until you learn the difference between phyla, classes etc. So learning the important differences between the members within an intermediate rank, e.g., how to separate the families within an order, is a very useful skill to have, and one which taxonomists will regularly exercise. (of course, there are still cases where even IDing something to family might also require a key!)
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They understand how to use keys, and how to recognise and interpret the characters and morphologies referred to in them. It’s one thing to read a key and understand that you need to appraise characters X and Y to identify a specimen, but it’s another to be able to look at said specimen and know where character X is and what character Y is. Morphological structures are often similar across related species of course, but there is a great deal of variation out there in many groups! Consider this example I came across recently.
This is a cool plant called Guichenotia intermedia from Western Australia. I spent ages going through the key, getting frustrated over the dead ends I kept reaching and not being able to figure out why I couldn’t get anywhere. In particular, all of the descriptions of petal size and colour for the species in Western Australia did not match my photos at all, even though I was very sure I had the genus correct. Then after reading a description of the genus more broadly, I realised my mistake; those giant pink lobes that look like petals aren’t actually petals! They’re actually the calyx, and the petals are the tiny red scales towards the centre of the flower.
So understanding the morphology of your group of interest, and what these structures actually look like, including how they differ between genera or species, is integral to key use. If an expert is using a key and gets to a step asking about hair type, they’ll know the difference between simple vs glandular hairs, or antrorse vs retrorse hairs. It’s knowing these characters, and what the different character states actually mean and look like in their taxon, that is a useful skill.
So long answer short, understand how to make identifications within your broad group at two different levels: a high level at coarse taxonomic ranks (I see a beetle, ok I know this is a ladybird, or I know this is a weevil, or I know this is a rove beetle), and a low level at finer taxonomic ranks (ok now that I know it’s a ladybird, I know to use this particular key, and when looking at the specimen in front of me, I know where the pronotum is, what the tarsus is, what the different terminology referring to the elytra means). If you apply these principles, it will certainly help you on your journey :)
I’m not a professional taxonomist, but have done a fair bit of work with a couple of insect taxonomists as an intern and volunteer, so hopefully my answer is a fair representation of just a couple of the many skills in a taxonomist’s toolbox (which will hopefully be expanded upon by the many amazing taxonomists that use iNat, especially since I haven’t even touched on actually describing new species!)