In Census 1911 searching the Brockville Census District, all the variations of Gallagher I can see live in the town of Brockville. In the Leeds Census District there are Gallaghers living at Newboro Village, Bastard and Burgess South Twp., and Kitley Twp.
In 1921 the Brockville census district (including Elizabethtown, and Brockville, etc) no longer exists and is enumerated as part of the Leeds District.
I have skimmed through a number of the 1921 sub-districts looking for any Gallaghers, these include:
SD 1-6, Bastard & South Burgess Twp.
SD 12-22, Elizabethtown Twp.
SD 24-27, Kitley Twp.
SD 49-63, Brockville (Town)
SD 71, Newboro (Village)
I found only:
1. John Gallagher in the Judson household (SD 26, page 4)
2. Houshold of James Gallagher (SD26, page 8)
3. a Robert Gallagher living in the Goulette household (SD 57, Page 2)
4. The William Gallagher household already mentioned (SD 59, Page 1-2)
So there are a few possibilities (aside from me having overlookied an entry, which is of course a possibility):
1. The family did not live in Brockville but in a township nearby and used Brockville as a point of reference - while I have looked at some of them as detailed above, there are a lot more I did not examine.
2. The family had migrated away from Brockville before 1921.
3. The children were born under a different surname (i.e. they became Gallagher's through a re-marriage of their mother while they were children)
I also noticed that most of the Gallagher's appearing in the 1911 Census in Leeds were a little older than what I would expect for a family of 5 to have not been started. So it is possible the Gallagher's left Brockville BEFORE census 1911
Since you mentioned someone (I presume Frederick) died in Ottawa, I looked at Carleton County. for 1911 and found a houshold of Gallaghers at Ottawa East that may bear further examination, The family is: Albertine b.1871, Frank b.1896, Mabel b. 1898, Beryl b.1901, Olga b.1905, Laura b.1909. Unfortunately they are no longer at this address in the 1921 Census, nor at Rosedale Ave., where the City Directory lists Albertina Gallagher (Wid Frank) living in 1923. I did not look at Lyon St, where the 1915 Deirectory lists her.