Your rationale is correct.
The relevant (in this case) simplified criteria for the choice of a principal chain in such compounds are:
- greater number of substituents corresponding to the suffix (principal characteristic group)
- longest chain
- lower locants for suffixes
- greater number of prefixes (e.g. “chloro-” or “hydroxy-”)
- lower locants for prefixes
- lower locants for substituents cited first as a prefix in the name
The corresponding wording of the nomenclature rules (taken from Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book)) reads as follows:
P-44.1 SENIORITY ORDER FOR PARENT STRUCTURES
When there is a choice, the senior parent structure is chosen by applying the following criteria, in order, until a decision is reached. These criteria must always be applied before those applicable to rings and ring systems (see P-44.2) and to chains (see P-44.3). Then criteria applicable to both chains and rings or ring systems given in P-44.4 are considered.
P-44.1.1 The senior parent structure has the maximum number of substituents corresponding to the principal characteristic group (suffix) or senior parent hydride in accord with the seniority of classes (P-41) and the seniority of suffixes (P-43).
(…)
P-44.3.2 The principal chain has the greater number of skeletal atoms [criterion (b) in P-44.3].
(…)
P-44.4.1 If the criteria of P-44.1 through P-44.3, where applicable, do not effect a choice of a senior parent structure, the following criteria are applied successively until there are no alternatives remaining. These criteria are illustrated in P-44.4.1.1 through P-44.4.1.12.
The senior ring, ring system, or principal chain:
(…)
(h) has the lower locant for an attached group expressed as a suffix (P-44.4.1.8);
(…)
P-45.2.1 The preferred IUPAC name is based on the senior parent structure that has the maximum number of substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
P-45.2.2 The preferred IUPAC name is based on the senior parent structure that has the lower locant or set of locants for substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
P-45.2.3 The preferred IUPAC name is based on the senior parent structure that has the lower locant or set of locants for substituents cited as prefixes to the parent structure (other than ‘hydro/dehydro’ prefixes) in their order of citation in the name.
You have correctly identified the principal characteristic group as carboxylic acid.
According to Rule P-44.1.1, the principal chain has to include the carboxylic acid. Therefore, the option labelled “7 carbon chain” can be ruled out. However, since there is only one carboxylic acid group in the given compound, this rule doesn't help you any further.
The next relevant rule is P-44.3.2, i.e. the principal chain is the longest chain. Therefore, the option labelled “6 carbon chain” is the correct choice. The resulting name is 2-(2-hydroxyethyl)-4-methylhexanoic acid.
Only if there is still a choice, the further rules would be considered. Thus, the mentioned “maximum possible number of functional groups” (other than the principal characteristic group) would be considered later in Rule P-45.2.1. Even in this case, however, the given explanation would not help since both names “2-(2-hydroxyethyl)-4-methylhexanoic acid” as well as “4-hydroxy-2-(2-methylbutyl)butanoic acid” have two substituents cited as prefixes.
