I'm finding that I don't like the word "subordinationism". There are better words to describe the belief that we're talking about. However...that appears to be the "word of the day".

Craig Keener (an egalitarian) wrote a rather long article: "Is subordination within the Trinity really heresy? A study of John 5:18 in context."
In the opening page he writes:

Nor, in fact, do Christological views coincide as closely with views on gender roles as some of the advocates of either position claim. Thus, for example, I frequently talk with Christians who espouse a complementarian view of gender roles while expressing surprise that anyone would deny the full equality in all respects of the Father and the Son. By contrast, I and some other scholars I know who support a very broad range of women's ministry affirm the Son's subordination to the Father. To be sure, that subordination may be voluntary, and we do not draw from it the same conclusions many of our complementarian colleagues do; but the fact remains that one's view on gender roles does not enable one to predict one's view of relations within the Trinity, or vice-versa. I do see evidence for the Son's subordination to the Father in rank; I also believe that evangelicals who differ on the matter should do so charitably. (emphasis mine)

The article begins at John 5:18

This was why the Jews were seeking all the more to kill him, because not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal with God. (ESV) (emphasis mine)

(A) Does Jesus Claim "Equality"? (5:18)

Jesus is "God the Son", but He is also acting as an agent for the Father. Keener makes the point that when we say that Christ as claiming equality with the Father in this passage, we are following the logic of Christ's enemies, not the actual words of Christ. Yes...clearly Christ is communicating His deity in this passage, but equality of roles with the Father? Keener believes not:

But while Jesus claims deity at various points in this gospel (e.g., 8:58; 20:2829), he also denies equality of rank with his Father. This is particularly clear in his response to those who think he has claimed such equality (5:19-30). Jesus does this by calling attention to his role as Son and agent. (emphasis mine).

In verses 19-23 we see the following points

• Jesus is following the example of His Father
• Jesus is saying that He can do nothing of His own accord
• Jesus has been given authority by His Father

Nowhere in this passage does Christ claim equality - He claims Sonship, with delegated authority and obedience.

(B) Jesus as God Son

Keener brings up a point that I had not heard of or thought of. Jesus was obediently following His Father's example. In the Jewish culture, how did a son learn his trade? By following his father's example - apprenticeship.

Nevertheless, this part of the discourse is framed with Jesus' claim not to act "from himself," or on his own initiative or authority (5:19, 30),25 fitting the Jewish conception of the agent who carries out his commission? Jesus elsewhere emphasizes that he does nothing "from himself" (5:30; 7:17-18, 28; 8:28, 42; 14:10), as the Spirit does not (16:13), and that the disciples cannot produce anything profitable from themselves (15:5).

(c) Jesus as God's Agent

In this section, Keener touches on the argument that yes-Christ was subordinate for the duration of His incarnation. But Keener points out that since Christ was "sent", that the submission started (at least) a little while before His birth.

Also, as a "representative agent" He carried the full authority of the Sender. This was in accordance with the time;

Agency represented commission and authorization, the sense of the concept which provides a broad conceptual background for early Christian agency. In many cases, at least in our later sources, the agent's own legal status was comparatively low. Indeed, under rabbinic rulings, even slaves were permitted to fill the position.32 Yet agents bore representative authority, because they acted on the authority of the one who sent them. Thus perhaps the most common rabbinic maxim concerning a person's agent is that "he is equivalent to the person himself."33 In the broader Mediterranean world envoys or messengers were backed by the full authority of those they represented. (...)

Even when one sent one's son (Mark 12:6), the messenger position was necessarily one of subordination to the sender. Although the concept of agency implies subordination, it also stresses Jesus' functional equality with the Father in terms of humanity's required response: he must be honored and believed in the same way as must be the Father whose representative he is (e.g., Tohn 5:23; 6:29).

(and I'm just a third of the way through the article...)

We have the framework for Christ's submission, obedience, subordination, and agency for the duration of His ministry while He walked this planet - and (at least a little) prior to.
Still...that does not provide proof that this submission is eternal.

Next up: section II: 1 Corinthians 15:28.