Can long range Wi-Fi work if one end of the connection is not using a high-gain antenna?

The antenna changes the shape of the transmission. The electrical signal doesn't get more powerful, but less is wasted in transmission in directions which aren't useful (e.g., up and down).

Similarly with reception, the signals are received from a more narrow field, this strengthens reception and reduces interference.

It's similar to talking through a cone, then listening through the cone for a response. the person at the other end doesn't need any special equipment, but you've increased your range and sensitivity.

"High gain antennas" provide gain on both transmit and receive.

So, with such an antenna at only one end, you'll get more range than with a standard antenna at each end, but less than with a high gain antenna at both ends.

Some things to consider: The wireless access point (WAP) receive side has one obstacle the transmit does not, and that is receive noise. The received signal will have a certain signal to noise ratio, i.e. how much above the noise the signal strength will be in db. The "gain" antenna can pick up more noise, especially if there is a "noise source" in the path between the two access points. Signal-to-noise ratio will directly affect the Bit Error Rate (BER) and throughput of the system. But, generally, focusing the transmit and receive signal in one direction with the antenna will increase the distance while maintaining the same BER, within limits.

The omni-directional antenna achieves "gain" by flattening the "donut" pattern so more signal is radiated in a straight lines, like extended wheel spokes, vice wasted with upward direction. This is achieved with the physical construction of the antenna.

Positioning two omni-directional antennas (whips) next to each other with proper "feed", at a calculated distance can also increase the "gain" bi-directionally.

Reflection of the signal from the ground, buildings and other objects may cause the direct signal and the reflected signal to "subtract" as they arrive at the wap receiver (fading). This is exacerbated with an omni-directional antenna.