Stationary ships (eg ... admirals extoll'd for standing still, Of doing nothing with a deal of skill) were used by the British fleet(s) off Ushant, Finisterre and elsewhere along the French Coast during the Napoleonic War, as part of the port blockade. However the practical gap was only about 10 nautical miles IIRC. This was only maintainable during good weather, as storms blew all the ships off their posts.
Note that this required large numbers of expensive, sea-worthy ships (frigates and ships-of-the-line), well crewed and officered, and easily supplied by remaining close to ports with fresh water and food.
Also, the messages that could be efficiently sent using signalling flags was quite limited. The straight-forward message England expects that every man will do his duty, sent by Nelson before Trafalgar, required 12 lifts and would have taken several minutes to send. Note that the flags of each lift must be raised, dropped, and of course left flying long enough to be read.
A similar system was used by fleets searching for another, as by Nelson searching for Napoleon's fleet before the Battle of the Nile in 1800. A screen composed of all small ships of the fleet would be thrown forward in an inverted V, so that all vessels crossing the path of the screen would promptly be made known to Fleet Command. This task is what the flag system was primarily designed to accomplish.
In the final analysis, one must ask what benefit would be accrued by the massive expense of a trans-Atlantic signalling fleet? It's size would have dwarfed even the Royal Navy of the 19th Century, and the messages capable of being sent more efficiently than by packet rather uncommon. What events, besides unfriendly Declarations of War occurring only a few times that century,
could have even approached being of sufficient value?
By my count the Royal navy employed 219 ships of the line and roughly an equal number of frigates during the span of the Napoleonic wars. With a distance of 3519 statute miles from Bristol to New York, a distance between stations of 10 statute miles on average, and an assumption that two ships would have been needed per station to allow adequate resupply, the entire Napoleonic era Royal Navy would have had insufficient ships (~700 required, ~450 available) to maintain such a telegraph, if it performed no other role. Clearly the astronomical cost of such a telegraph made in unfeasible to even consider implementing.
As for pigeons, the world record for a homing flight is from Arras, France, to Saigon, Vietnam in 24 days. A distance of ~7200 miles, that gives an average speed of 300 miles per day; exactly the same expectation as for a ship-of-the-line crossing the North Atlantic. Where pigeons had an advantage was over land rather than water, where daily travel distances were typically much less.