According to Star-K article, no:
Since many people enjoy their tea mainly on Shabbos, it would be
appropriate to briefly mention the do’s and don’ts of tea preparation
on Shabbos. One should not use tea/herbal bags or loose teas on
Shabbos. This is because tea is part of that group of foods known as
kaley habishul, or easily cooked foods. These foods are considered so
sensitive to heat since they will cook in circumstances that other
foods will not.
Therefore, the only way to enjoy tea on Shabbos is to prepare tea
essence before Shabbos. The common practice is to make a very strong
cup of tea before Shabbos by using several bags and pouring boiling
water over them. It is preferable to also remove the bags before
Shabbos. This essence can now be added to a cup of hot water (kli
sheini) on Shabbos. The preferred method, however, according to Rabbi
Moshe Heinemann, shlita, Rabbinic Administrator of the Star-K, is to
place several bags into a pot of water and actually boil them up on
the fire before Shabbos. This essence, minus the bags, can be stored
in a cup and added to hot water.
Alternatively, you may want to use instant tea on Shabbos. When using
instant tea, or coffee, the preferred method is to add the tea to the
hot water (kli sheini) and not vice versa.
See the footnotes in that article for sources.
I listed just one source. There are others that permit making tea using tea bags, and probably even using the loose tea in a cup.
However, I found no opinion that allowed use of the tea spoon as this is a special tool designated for straining, which is similar to the melacha of "borer".