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"William Brooks, a British man who lived in France during the 1800s, "

How do you know that? Do you have an address?

" Harry Brooks, born in London around 1864 (we believe his mother Eliza would’ve been ~34 at the time)"

So William was still in London, presumably, in about 1864?
 
   "Alice Brooks, born in Rocquencourt, France in 1889 (we’ve found her record already in the Yvelines Departmental Archives)"

Could she be Harry's child and William's grandchild?

I already noticed that Eliza would have been far too old to be her mother in 1889, never mind that you think William was dead in 1887 .
Presumably tha record you have found names the father, at least ? Please give the details or ask for them to be transcribed/translated.

"But does anyone have any suggestions as to why the place was stated for this baptism, but not for any others on this page?"

Because the other entries refer to families who lived in the parish of Uttoxeter whereas this child was born elsewhere, perhaps in the grandparent's native hamlet or village? 8 miles was not far to walk in the 1600s !
Or perhaps the other way round: the parents lived in Wood-end and the grandparents (where the child was born) lived in Uttoxeter. The child was a bit sickly and was baptised within a few days?

Here are the Jeffery and Symes records I found in the Brent Knoll (formerly South Brent) registers on Ancestry. If you have a subscription you can check them yourself, of course, by browsing the Brent Knoll 1531-1800 register. I may well have missed some:

Baptisms
1788 March 16 John, son of Joel and Mary Symes
1789 Oct 26 William,  ..         ..          ..        ..
1791 Sept 18 Mary, dr   ..     ..        ..            ..
1793 July 28 Sarah, dr ..     ..        ..             ..
1795 May 31 Petter  sn   ..    ..      ..             ..

Burials
1771 June 28 Mr Edward Symes, Gent
1781 Edward symes surgeon (??) buried
1784 Nov 1 Betty Symes died
1794 June 9 John Symes, Gent was buried , died June 3
1796 Sarah, dr of Joel & Mary Symes buried
1797 Jane Symes widow buried
1798 Dec 22 Elizabeth Symes widow buried
1799 Sept 5 Elizabeth Symes buried

Jeffery
1681 Nov 1 William, son of William Jeffery baptised
1691 Oct 13 John son of Thomas & Elizabeth Jefferies baptised
1692?? Maud, wife of William Jeffery, yeoman, buried
1704 Jan 19 Joan?? Jeffery widdow buried
1704 Nov 10 Thomas son of Thomas & Elizabeth Jeffery buried
1704 Aug 22 William son of William & Elizabeth Jeffery baptised; buried Jan 12 1705
1705 July 13 Thomas son of Thomas & Elizabeth Jeffery baptised; buried Oct 14 1705
1752 June 1 William of Mark and Joan Jeffery of Burnham married

Hornett records:
1784 July 4 Elizabeth dr of Robert & Martha Hornet baptised
1785 Nov 6 Ann        ..         ..           ..          ..        ..
1789 Elizabeth Hornet was buried
1790 Oct 11 John     son    Robert & Martha Hornet baptised
1793 May 21 Isaac     ..       ..         ..             ..           ..
1796 Aug 7  Elizabeth  dr     ..         ..              ..           ..
1799  April 7 Robert  son     ..         ..               ..            ..

Chard records:
1787 April 15 Betty dr of Richard Chard baptised
1797 Jan 17 Betsey dr          ..    and Sarah Chard baptised
1800 Jan 5 William son of Richard & Betty Chard baptised

No sign of Faustin Jeffery I'm afraid.
Hope it helps.
 

I don't know if the OP is still with us but I've just come across the baptism of a William Chard, son of Richard and Betty Chard in Brent Knoll on July 29th 1792 (Ancestry image).

Also numerous children baptised to Joel and Mary Symes in the 1780s and 1790s - in case that information is of any use to dismo and his search for Jeffery/Symes connections.