All other forms of verbs inherently seem to have some kind of meaning but te needs to be combined with things before it means anything. Does it mean anything on its own that makes what it ends up doing with other things make logical sense? Like, why do te kudasai and te iru/aru both use te?
In your examples, both the words following te are verbs and te are supposed to be attaching verb phrases before them. 助詞(joshi)/particles(to which te belongs) are grammatical tools and don't have meanings on their own. A meaning is made when they are attached to other words and how they are constructed.
Of course, there are exclusions in idiomatic expressions like "で？ (何？)", but these should construed as a shortened form of a phrase.
"te" by itself have no meaning. There is canonical word of verbs and if You add te in correct way it changes verb meaning. kaku - to write (canonical) kaite - write (in "You write", used with friends, family, coworkers,...) kaite kudasai - please write (more official form, used with people You don't know well)
Let me divide your question into two separate questions...
All other forms of verbs inherently seem to have some kind of meaning but te needs to be combined with things before it means anything. Does it mean anything on its own that makes what it ends up doing with other things make logical sense?
When spoken, the "te" form of the verb (by itself, with nothing following it) means the same as "te-kudasai":
- 電話してください（Denwa shite kudasai）(Please call me.)
- 電話して（Denwa shite）(Call me.)
Using "kudasai" is much more polite, but the meaning is the same. You will never see the "te" form by itself in written Japanese (with nothing following it) except if the text is quoting spoken Japanese.
Like, why do te kudasai and te iru/aru both use te?
A verb's "te" form allows it to be combined with other verbs to either conjugate the original verb (as in "te-iru" and "te-aru") or to sequentially combine the verbs (e.g. "itte-kuru" means "go then come back"). Here "te" plays a purely grammatical role and has no meaning by itself.
Note that "te-kudasai" is a combination of one verb's "te" form and the imperative form of the verb "kudasaru".