grammar - Should I use ～てきた or ～て来た?

In the examples below

彼は私のところに荷物を送ってきた。
彼は私のところに荷物を送って来た。

荷物が私のところに送られてきた。
荷物が私のところに送られて来た。

would it be better to use the termination/set phrase てきた？ (no kanji) because this is changing the nuance of the sentence? Or should it be understood literally (arrived) and therefore we て来た (with kanji) is more natural?

Answer

TL;DR: Whichever is fine, but using くる is safer.

---

You seem to know the basic rule, but just to make sure, this type of verb after te-form of another verb + て/で is called a subsidiary verb, and they are generally written in hiragana.

But this rule is not very strict, and every so often you will see people use kanji for subsidiary verbs, especially in stiff and stilted writings.

However, verbs after the te-forms are not always subsidiary verbs. きた in 暖かくなってきた is definitely a subsidiary verb, while 来た in 彼は急いで来た ("He came in a hurry") is a normal verb, so it is usually written in kanji. 見た in 箱を開けて見た ("I opened the box and looked (inside it)") is used as a normal verb, while みた in 箱を開けてみた ("I tried opening the box") is used as a subsidiary verb.

So the problem is this: Is きた in 荷物を送ってきた a subsidiary verb or a normal verb? How about きた in 荷物が送られてきた?

Well, I believe these きた can safely be considered as a subsidiary verb. デジタル大辞泉 clearly defines these usages under the 補助動詞 section:

９ （補助動詞）動詞の連用形に接続助詞「て」が付いた形に付く。
(snip)
㋒ある動作をしてもとに戻る。…しに行って帰る。「買い物に行ってくる」「外国の事情をつぶさに見てこようと思っている」

㋓ある動作・状態をそのまま続けながら、こちらへ近づく。また、そのようにしてこちらへ至る。「敵が押し寄せてくる」「付き添ってくる」

So the safer choice is to use hiragana for both of your examples.

But I also feel using kanji is not so unnatural in these cases, because something is physically moving. People often use the kanji 来る even in casual situations when something is physically moving to you. I personally am inclined to use hiragana for those cases, but perhaps this is a gray-area matter and it's up to the writer's taste.