No. Gearbox oil (or lack of it) does not stop the gears engaging. It only prevents the gearbox components wearing out.
Garages always suggest "change the gear oil" as a catch-all suggestion but I've lost track of the number of times an OP writes... "....the garage changed the gearbox oil...." and "... it was better for a while..." and ".. but now it's the same as it was before..." all in the same post.
You're right. However the slave cylinder is sitting on top of the gearbox, which is hot.. so it will get warm and that will affect the push-rod and the seals. Even without a slave, the internal gearbox/clutch parts do get hot so the clearances within the clutch and gearbox are different when it's all warmed up, which might affect how easy the gears are to shift.
Yes it could.. Worn out clutch plates are more likely to have problems not slipping, rather than not disengaging the drive... but.. the slave activates the clutch by rotating a forked arm inside the clutch bell-housing that presses a release bearing against the clutch pressure plate, which disengages the clutch. After a while the bearing body starts to wear a groove into the pressure plate fingers, so that when the forked arm pushes it, some of the movement is not transferred to the clutch pressure plate because the groove introduces "slack" into a chain of what should be just/barely touching parts.
You can also get wear of the forked arm and/or wear of the bearing body itself.. but the bearing and pressure plate fingers are the softer parts so they're more likely to wear out... They both come in the same clutch kit, so there's no point trying to work out which one of them is the main culprit.. by the time you get this far in the dismantling, just replace both of them regardless.
The best clue to the cause is to listen to the engine while it's idling. Get your head under the bonnet.. If you can hear a scraping sound like shhhick shhhick, shhhick... etc. get someone to press the clutch pedal in. If the noise quietens down, then that's the release bearing (you will need a new clutch kit). If the gearbox/clutch are relatively silent and don't make any more or less noise when the pedal is pressed, then it's more likely to be the slave cylinder.
Ralf S.