To ask an analogous question about the shapes of the letters in the Latin alphabet:
Does the letter "E" contain the the letter "F"?
It seems that "E" is just "F" with an extra horizontal line on the bottom.
In terms of shapes in the modern script, it is trivially obvious that "E" contains the strokes of "F". What "E" does not contain is any functionality of "F"; the sound of "F" does not contribute anything to the sound of "E".
"E" containing the strokes of "F" does not explain any functional relation between "E" and "F", if there is any at all. Similarly,「犬」containing the strokes of「大」provides no explanation of the functional relationship between those two, and this is what you're missing.
For reference, "E" and "F" are unrelated. From Wikipedia:
The Latin letter 'E' differs little from its source, the Greek letter epsilon, 'Ε'. This in turn comes from the Semitic letter hê, which has been suggested to have started as a praying or calling human figure (hillul 'jubilation'), and was probably based on a similar Egyptian hieroglyph that indicated a different pronunciation. In Semitic, the letter represented /h/(and /e/ in foreign words); in Greek, hê became the letter epsilon, used to represent /e/. The various forms of the Old Italic script and the Latin alphabet followed this usage.
The origin of 'F' is the Semitic letter vâv (or waw) that represented a sound like /v/ or /w/. Graphically it originally probably depicted either a hook or a club. It may have been based on a comparable Egyptian hieroglyph such as that which represented the word mace (transliterated as ḥ(dj)):
The Phoenician form of the letter was adopted into Greek as a vowel, upsilon (which resembled its descendant 'Y' but was also the ancestor of the Roman letters 'U', 'V', and 'W'); and, with another form, as a consonant, digamma, which indicated the pronunciation /w/, as in Phoenician. Latin 'F,' despite being pronounced differently, is ultimately descended from digamma and closely resembles it in form.
「犬」and「大」are similarly unrelated, but you wouldn't be able to immediately know this without exploring a bit of history. Their shapes converged because of the way writing was stylised and streamlined.
「犬」was originally a picture of a dog.
「大」was originally a frontal view of an adult person, indicating the meaning adult > big, large. Choosing samples from roughly the same location and period in time, the comparable development looks like
Of course, there are modern shapes containing the strokes of「大」which really did contain「大」historically, such as「因」.
There are also things that originally contained「大」, but which is not recognisable in the modern script anymore, such as「達」, where「大」is squashed into a shape identical to「土」.
「大」would have otherwise been more obvious using the oracle bone form: