Beyond all certainty, the answer is 'yes'.
Sputnik 1 was specifically designed as a propaganda ploy, the singular intention of which was to broadcast a radio 'beep' that could be picked up on regular AM radios. There was no other practical purpose of the ball. That's how we (by 'we', I mean my family and I, personally) detected it as it flew overhead. Of course, we were able to see it as well, given the time of night and the position of the sun reflecting from it.
Sputnik 1 was less than 23 inches (58 cm) in diameter in an orbit between 215 km. and 939 km. With the state of radar at the time, it would be undetectable by any Earth-based system unless they were specifically looking for it.
It wasn't until 1960 that America had a radar system sensitive enough to pick up satellites that did not emit radio signals.
Three weeks ago, headlines announced that the U.S. had detected a mysterious "dark" satellite wheeling overhead on a regular orbit. There was nervous speculation that it might be a surveillance satellite launched by the Russians, and it brought the uneasy sensation that the U.S. did not know what was going on over its own head. But last week the Department of Defense proudly announced that the satellite had been identified. It was a space derelict, the remains of an Air Force Discoverer satellite that had gone astray. The dark satellite was the first object to demonstrate the effectiveness of the U.S.'s new watch on space. And the three-week time lag in identification was proof that the system still lacks full coordination and that some bugs still have to be ironed out.
First Sighting. The most important component of the space watch went
into operation about six months ago with the construction of "Dark
Fence," a kind of radar trip wire stretching across the width of the
U.S. Designed by the Naval Research Laboratory to keep track of
satellites whose radios are silent, it is a notable improvement on
other radars, which have difficulty finding a small satellite unless
they know where to look. Big, 50-kw. transmitters were established at
Gila River, near Phoenix, Ariz, and Jordan Lake, Ala., spraying radio
waves upward in the shape of open fans. Some 250 miles on either side,
receiving stations pick up signals that bounce off any object passing
through the fans. By a kind of triangulation, the operators can make
rough estimates of the object's speed, distance and course.
On Jan. 31 Dark Fence detected two passes of what seemed to be an
unknown space object. After detecting several passes during the
following days, Captain W. E. Berg, commanding officer of Dark Fence,
decided that something was circling overhead on a roughly polar orbit.
He raced to the Pentagon and in person reported the menacing stranger
to Chief of Naval Operations Arleigh Burke. Within minutes the news
was communicated to President Eisenhower and marked top secret.
But it begs the question be asked, 'Why would the Russians send up a satellite no one could detect, if the entire purpose of the launch was to prove to the Americans that Russia could do it?' It was only after this, that they sent up silent spy satellites that took high-res photographs, and then were retrieved to get the film.
So, you think taking your film to the local shop to get developed is a
pain? Try being an American spy satellite in the 1960s. Getting your
film developed then meant dropping it in a special ‘film bucket’
capsule from space, which the US Air Force then had to catch in
mid-air.
Strange as this seems, this is in fact how it worked, as you can see
in the video above. Photographs captured by these so-called “Corona”
satellites were shot on special 70 millimeter Kodak film using two
panoramic cameras that evolved over the course of the program.
From Wikipedia Note this refers to the booster, not Sputnik itself.
The booster rocket was located and tracked by the British using the
Lovell Telescope at the Jodrell Bank Observatory, the only telescope
in the world able to do so by radar.[48] Canada's Newbrook Observatory
was the first facility in North America to photograph Sputnik 1.[49]
Edit Addendum
Sputnik 1 was the first, period.
There were no satellites before Sputnik.
There were no ICBMs before Sputnik.
There were no rockets capable of delivering a payload into orbit before Sputnik.
There was nothing, period, made by humans circling the Earth before Sputnik. There was zero capacity for any country to put anything into Earth orbit.
There was certainly absolutely no need for anyone to have any radar that was looking for anything orbiting the Earth. It just wasn't possible for anything to be there.
For anyone under the age of 70, it is very difficult to imagine a world when there was absolutely nothing made by humans orbiting or even capable of orbiting the Earth. But some of us are old enough to remember. We were alive when the only thing in space before Sputnik was in sci-fi novels and comic books.
And there was zero telemetry from Sputnik 1. All it did was beep. The only telemetry was a change in the frequency of the beep, depending on temperature and pressure - catastrophic failure of the vehicle. No other data was transmitted. The only purpose of the beep was to demonstrate it was in orbit. When they heard the beep 90 minutes later, over Russia, they new it was a success. End of story, end of data, end of reason for the beep except propaganda.
EDIT Addendum 2
America did not even have radar installations that would reach near-Earth orbits, nor that even covered their own air space, until after 1957.
The new radars will help CAA controllers accomplish this by scanning
the skies for all aircraft up to 200 miles away, depending on size and
altitude.
The 23 radars will be part of an expanding coast-to-coast traffic
control network of more than 70 civil and military radar
installations. The network will give controllers a picture of aircraft
from 15,000 to 70,000 feet in virtually all the U. S. airspace, and of
aircraft at lower altitudes on densely traveled routes. Thus, radar
will serve to track the civil and military jets which move at 600
miles an hour or more in the higher altitudes, and the conventional
aircraft traffic using the lower altitudes.
EDIT Addendum 3
The irony is, it was the Russian push for a propaganda victory that led to the Americans winning the race. Had the Soviets kept it a secret, world history would have been very different. America would not have changed their educational system towards the '60's version of STEM, physics would not have gotten the boost it did (PSSC Physics textbooks re-wrote the entire physics curriculum), America would probably not have put huge resources into space radar, and Kennedy would never have pushed for 'the Moon within a decade'. There would have been no Apollo program, no moon landing.
As for the timing deice, it was well within electronics/electricity to make a very simple capacitive timing device using diodes and resistors, to cause a capacitive discharge into a coil to produce a spark discharge every 90 minutes (the length of time it took for one orbit). The spark would have been so broad-band, it would have been picked up by receivers around the world, but would have been impossible to completely localize geographically, let alone be localized to space. The West would know something somewhere in Russia caused a spark, but they would have no idea what, and would probably not be able to differentiate it from say lightning. The Russians would know Sputnik completed an orbit, but no one else would have a clue. Recall that the Russians had absolutely no way to track it by radar either. They too needed it to send a signal.
The Russians could have developed their ICBM project in secrecy. They should just have kept their mouths shut, instead of going for publicity.
