Well, usually they don't got to a planetary system cold turkey.
So, lets say for an example a standard planetary survey on your 5 year mission:
some type of probe has probably been through the area before, maybe decades before, so you start with that information, is the system inhabbited, does the planets have anything of interest etc.
Now you have alot of different departments that would read the reports, and then go through the new sensor readings.
Exo Biology would see if there was any life in system
Stellar cartography would see if there were any changes in the system, sun is healthy, etc.
If a warp drive was detected, a first contact specialist would study the culture etc. of the planet.
Etc. Etc.
and all this would be written in a report, and given to the chief of operations, who would be on the bridge during the entry of the system, and they would pass on any information to the captain of note, and probably post the reports up for any crew memeber to read.
So during the entry to the system, you wouldn't have a geologist or xeno biologist on the bridge, you'd have the department chiefs at stations.
Now if there was anything out of the ordinary, like First Contact for example.
The captain would know alot from the reports, but a First Contact specialist would have read up on the planet, and know all the customs, etc. of the society. This would be true on almost any contact, doesn't have to be first contact.
Say there going to visit a society that they have already contacted before, you would still have a contact specialist that would read up on the society, and report to the captain anything they should know, and they might even be on the bridge if the contact was out of the ordinary.
But most times, your standard bridge crew would be on the bridge, making decisions, advising the captain of what to do based off of reports from everyone down the line before hand.
Now in TOS we had Botinists, Geologists, etc. on away missions, TNG almost always just had Data for all the science, so we never seen any geologists on the D. Now, I know real world reasons of, they need to hire somone to play said geologist, if they speak, you have to pay them. Your already paying Data to do the science, so why hire some guest to fill out the away team.
But in answering the original question.. Yes, there should be more specialists shown doing there work, not just the main cast.