The dating is not used to define the temperature or other chemistry. It is used to put the sample into a time context. Most dating using radioactive decay as a "stopwatch" has a window of time where it works, and only works if the radioactive element and its daughters get captured at the time the mineral, rock, or organic residual got made, and any changes after formation to concentration of those atom types in that sample is only because of decay).

The date we would get from measurement and calculation refers to the date that the measured component got created. Dating the age of a mineral derived from erosion of an igneous rock will yield the age of the igneous rock, not the age of the sediment, so sediments tend to be more difficult to date by radioactive means. Some minerals do form in the sediments at (about) the time of sediment deposition, and dating those minerals, when possible, will give a decent age for the sedimentation. Generally speaking, radioactive dating is used to put date limits on sequences (using some cross-cutting relationships with datable units like dikes or sills; younger than that 5 million year old rock but older than this 3.2 million year old rock). Ash layers from volcano eruptions are actually really good for that, because the ash layer is a distinct layer and a fixed time and covers a very large area, and the age of the ash is the age of the sediment it is in, so it can be a very good marker unit for a lot of different places in the area.

Some fossils are actually pretty good for dating, but short time windows tend to rely on microscopic fossils like foramnifera that are a lot of work to find and ID). Carbon dating (which only works on fairly young materials, like 60,000 years old or less, about) isn't actually dating the sediment either, but the age of the piece of wood or plant leaf or whatever is going to be the same (to the precision we can measure) as the sediments it got buried by and within.

Mostly, though, we either use other non-radioactive methods to date the sediment or material. Ice ores are dated sort of the same way as trees using tree rings. The rings (layers) get counted. It isn't always counting right from surface, sometimes the section or the sample is compared to other sequences already counted and matched to a window of time without counting all the way back until today in that one sampling program.

The point here is that dating is not the same analysis as the measurements used to define chemical conditions of atmosphere and oceans. There are lots of ways to measure temperature and chemistry, some of which are very direct (when you can get an actual, real sample of preserved atmosphere in some ice and analyze it) and some are indirect (using stable isotope equilibrium among different species to define the temperature of equilibration (=formation) assume the minerals formed from sea water did so in equilibrium with that sea water and thus with each other). Similar things respecting ocean chemistry is revealed by major and trace element contents of minerals formed from that sea water. So we can get a pretty decent idea, within certain windows or ranges (small error limits), about what ocean chemistry and temperature was at the time of the deposition of sediments in that ocean.

It isn't just one thing and "POOF" the conditions are known for a given time. It is a lot of work involving measurement from a lot of samples from a lot of different places and times, and fitting all the data into how things have changed (or stayed the same) through time.

We keep doing sampling and analysis of different sediments, or ice, or whatever we can get our hands on, from different places and different times, and adding that data to what we already know, and this allows us to know more precisely, better, what actually happened. The more information we have, the more certain we can be in our understanding of things. It takes time and lots of work by lots of different people though.