Because the IF carries the signal of interest, we do not want any of it to leak, that could require
further amplification to make up for the signal loss, and any amplification adds noise.
Also that if it can leak out, signals of the same frequency can creep in, further reducing the
sensitivity of the receiver.

So basically by design - we want to prevent ingress an egress of signals on the IF; we don't care so
much about the LO(s).

There will always be some possibility of IF ingressing / egressing the tuned circuit (as perfection is
not possible), but such signals should be significantly attentuated.

My thoughts were that the IF is processed by more electrical components and at higher power levels than the LO frequency, hence I'd expect the more powerful electromagnetic radiation (EMR) to be from the IF. Also, because the LO is at the initial stages of the signal processing, the screening of it would be more important to prevent degradation of the TV signal by noise at an early stage of the processing; the better the screening, the lower the EMR.

Err - not the way I'd approach it.

Yes more components may handle the IF (or IFs), but (in UHF gear) these components tend to be within shielded boxes (open a receiver and look for the metal boxes). The express purpose of these boxes is to prevent stray signals (at the IF) for entering the circuit. Said shielding also tends to be symmetrical - it also acts to attenuate any IF that leaks. The shielding also tends to be partitioned,
with different frequency ranges within each partition.

The LO may be within or without the shielded area. It will be operating from a local power supply, said supply will usually be at least 1 volt. The LO signal will tend to be generated in the range of milli volts to volts (assume mV).

The signal received from the antenna will tend to be at micro Volt levels (1000 times less than the assumed LO signal). That signal will be mixed with the LO to generate the IF, the signals may be set to similar levels before mixing, either by attenuating the LO, or by amplifying the antenna signal.

Given a sensitive enough detector in the demodulation state, the 'best' choice is to attenuate the LO, since any amplification adds noise, so we now have a low level IF signal.

How is the LO going to degrade the input signal?

The Q of the input circuit, together with the power levels fed in to the mixer will tend to prevent this.

Consider a single IF receiver.

Carrier at around about 500MHz (UHF UK TV), IF at around 50MHZ, this requires a LO of around about 450MHZ. This will give mixing products at 50MHz and 550MHz.

The LO and upper mixing product are close to the carrier, and will more easily leak out of the input tuned circuit (say it covers 300Mhz - 700Mhz). The IF however is way outside that range, so should tend not to leak.

Anyway, I tire of this.

Go and read up on how superhet receivers work, and mixing theory. Peruse a few data sheets for UHF receiver components.

Then disassemble a few receivers. For convenience pick a simple pocket VHF (FM) radio. Probe the circuit with a 'scope while tuned to say Radio 4, and note the signal levels at various frequencies. You should be able to determine the LO frequency without too much trouble.