And if it is possible to use a tuner to make a transmitter feed power into a coat hanger then why do we worry about making resonant antennas in the first place?
In a word - efficiency.
Consider that a full size 40 meter, 1/2 wavelength dipole is approximately 65 feet (19.8 meters) long. A dipole that is only 2 feet (0.6 meters) long, would have a gain of only 0.5 dB less than the 65 foot dipole. This would hardly be a noticeable difference on the air. But the problem is we cannot make the antenna system efficient enough to realize that minor 0.5 dB loss. The system inefficiencies often result in substantial losses compared to a full size dipole.
But we also should not be obsessed with resonant antennas - it makes little difference with practical antennas if we properly deal with their non-resonant effects. A 10/8 wave dipole has nearly double the gain of a 1/2 wave dipole. It is not resonant but can easily be matched to a transmission line with a simple matching network. A Yagi antenna is not resonant without the addition of a matching network. Also consider that many resonant antennas are not resonant at the impedance of the feedline anyway - for example, an ideal 1/4 wave ground plane vertical is resonant at approximately 34 ohms.
Here are some typical sources of antenna system inefficiencies:
Radiation Resistance vs Resistive Losses
Radiation resistance is an often misunderstood and misapplied term. The radiation resistance of an antenna is caused by the radiation of electromagnetic waves. For a free space, 1/2 wave, center fed dipole of reasonable construction, the radiation resistance is ~73 ohms. Any resistive losses (power that is not radiated) in the antenna add to this radiation resistance to contribute to the feed point impedance. In the case of this dipole, there will be a very small amount of RF resistance due to the wires that make up the dipole. If we use 14 gauge (1.45 mm) wire to construct the antenna, the RF resistance will be ~2.7 ohms. The total feed point impedance would then be 75.7 ohms in this example.
The efficiency of an antenna is given by the formula:
$$Efficiency = \left( \frac{R_r}{R_r+R_l} \right)$$
where Rr is the radiation resistance and Rl is the resistive losses.
So if we apply this to our dipole example above, the efficiency would be 96.4%. By contrast, the two foot dipole will have an Rr of approximately 0.04 ohms and a comparative Rl of 0.08 ohms resulting in an efficiency of 33%.
To complete this part of the efficiency picture, consider that:
$$ Gain = Directivity \times Efficiency $$
A full size dipole has a directivity of 1.65. Multiply this times the 96.4% efficiency of the above example and the gain becomes 1.59 (2.02 dBi). For the short dipole, the directivity is 1.5. Multiply this times the 33% efficiency of the above example and the gain becomes 0.5 (-3.01 dBi). So we already have a 5 dB difference between the two antennas and there still are other system losses that must be taken into account.
Matching Networks / Tuners
In the example of our 1/2 wave dipole, the feed point impedance of the antenna is ~ 75 ohms. If we are attempting to drive this with a 50 ohm source, we may wish to have a tuner or matching network that does the transformation between the two largely resistive impedances. In this case, a well built tuner or matching network will have an efficiency in the 80-95% range (less than 1 dB of loss).
In the case of our short dipole example, the situation is a bit more complex. We need to both cancel the capacitive reactance of the dipole and match the very low feed point impedance. Without going through laborious calculations of matching network efficiencies, it would not be unreasonable to expect less than 10% efficiency (>10 dB loss) from the matching network. This means that at least 90% of the applied power or signal will be lost to heat in the matching network alone. Our short dipole system now has a gain of -13 dBi which is more than 15 dB down from our full size dipole example. This is the equivalent to comparing the signal strength of a 100 watt transmitter to a 3 watt transmitter.
Transmission Line Loss
All real world transmission lines exhibit loss. This in itself is a reduction in efficiency. The specification for the transmission line lists the loss for the transmission line at a given frequency assuming that the transmission line is terminated in its characteristic impedance. When the transmission line is not terminated in its characteristic impedance, the loss of the transmission line increases (the efficiency further decreases).
By placing a tuner or matching network close to the antenna to provide a match to the characteristic impedance of the transmission line, the additional losses due to a transmission line mismatch can be avoided. This maximizes the efficiency of a given transmission line.
There is a secondary effect when the transmission line is not connected to a load that matches its characteristic impedance - the transmission line will no longer exhibit its characteristic impedance. Another way of saying this is it becomes an impedance transformer. For example, our 75 ohm dipole impedance in the early example when connected to 23 feet of RG213 will be transformed to 34 ohms at the transmitter end of the transmission line. The total losses in the transmission line will be 0.119 dB, of which only 0.009 dB is due to the mismatched load. A free program such as TLDetails makes quick work of these calculations:
In some cases, this transformation can be used to our advantage by transforming the antenna impedance to something that is more usable from a system efficiency perspective. In other cases, it can worsen the system efficiency. But do consider this transforming effect in light of transmitter output efficiency.
Transmitter Output Efficiency
The final efficiency factor to be considered is the output power of the transmitter. Most amateur transmitters are designed to deliver their rated output power when connected to a 50 ohm resistive load. Any deviation from this load will typically result in lower output power from the transmitter. This loss of transmitter output power effectively reduces the efficiency of the system.
For example, a 100 watt, 50 ohm source impedance transmitter that is connected to the 23 feet of RG213 that is terminated with our 75 ohm dipole, will output 96 watts (if no protection circuits kick in). This is an output efficiency of 96% (a 0.17 dB loss). Note that this is most likely more efficient, in this example, than using an antenna tuner at the transmitter to match the transmitter impedance to the feedline impedance (even though this is a 1.5:1 SWR).
SWR
SWR is based on the relationship of the transmission line characteristic impedance to the impedance of the load on the transmission line. In the case of a transmitting system, the load is typically the antenna. In the case of a receiving system, the load is typically the receiver input impedance.
Since we have already dealt with the relationship of the transmission line characteristic impedance to the load impedance in the earlier transmission line section, we need not consider SWR as another source of inefficiency.