I agree with Mick and Nick that what is the best approach to Theme and Rheme depends on what you want to do with it. Like them I’m particularly interested in patterns across texts.
It could be said that in English the main function of Theme is to show whether the perspective of the text changes or whether it remains the same, in particular whether the main topic entity changes or stays the same and whether the setting changes or stays the same.
In Mick’s example, the ‘He’s show that the writer is staying with the same main topic entity. But at intervals the temporal setting changes, as shown by the initial Adjuncts. As Mick has shown, for me both the continuity of main topic entity and the changes of setting are relevant to the ongoing perspective of the text. So I want the Subjects to count as Themes as well as the Adjuncts. (Which Halliday wouldn’t allow when a Subject is preceded by an Adjunct.)
 To be clear, if a theoretical term like 'Theme' is not used in the original formulation, the use of the term is no longer valid, since the theoretical valeur of the term has changed. Moreover, this can have unintended consequences for systemic relationships, the basis of explanation in SFL. Halliday & Matthiessen (2014: 49):
Giving priority to the view ‘from above’ means that the organising principle adopted is that of system: the grammar is seen as a network of interrelated meaningful choices. In other words, the dominant axis is the paradigmatic one: the fundamental components of the grammar are sets of mutually defining contrastive features. Explaining something consists not in stating how it is structured but in showing how it is related to other things: its pattern of systemic relationships, or agnateness (agnation…).
 To be clear, thematic patterns across texts can be examined using the original formulation of Theme. That is, this does not constitute an argument in support of varying the theoretical valeur of 'Theme'.
 To be clear, this characterisation of Theme confuses Theme, a functional element of the clause, with the logogenetic pattern known as the method of development. As Halliday & Matthiessen (2014: 89, 126) explain:
The Theme is the element that serves as the point of departure of the message; it is that which locates and orients the clause within its context. The speaker chooses the Theme as his or her point of departure to guide the addressee in developing an interpretation of the message; by making part of the message prominent as Theme, the speaker enables the addressee to process the message. The remainder of the message, the part in which the Theme is developed, is called in Prague school terminology the Rheme. …
The choice of clause Themes plays a fundamental part in the way discourse is organised; it is this, in fact, that constitutes what has been called the ‘method of development’ of the text…
 To be clear, this argues for a reformulation of a clause function (Theme), not on the basis of its function in the clause, but on the basis of logogenetic patterns, and in doing so, confuses two distinct logogenetic patterns: choice of Theme and choice of Subject.