"Zero-cross" switching is irrelevant, it's almost exclusively a marketing thing. It has no bearing on device reliability or whether or not it'll damage the telly. It merely reduces EMI switching noise - which is a good thing if you've got a lot of them, otherwise irrelevant.
- Consider a 3-pin-plug or the switch on the socket. Does that do zero-cross detect? Does the act of plugging in a TV damage it?
AC SSRs are actually a pair of back-to-back SCRs or a Triac, just like a dimmer. This means that the waveform they produce is not sinusoidal (both due to being non-resistive and having a recovery time during zero cross) and that they never fully "turn off" - there is always a small leakage current through the SSR. There is also a notable insertion loss (fixed voltage drop) which makes them rather inefficient compared to a relay.
Many power supplies can be damaged by these effects - continually feeding it a small current that's not enough to start it and making the zero-cross point fuzzy are both bad things to do to a power supply. Switch-mode PSUs also draw their current in 'pulses', which means the SSR may not actually stay turned on throughout the cycle and may 'starve' the SMP. (This has all kinds of strange effects)
Put simply, you need a physical relay contact that goes "click", because that's what the PSU in your equipment was designed for.
The best is a physically latching one as then no power is consumed keeping the relay "on" - this is impossible to do with SSRs as they must be powered to turn on. Thus relay-based ones can be considerably more efficient than SSR ones.
That's assuming you even come out ahead in the first place - if you have multiple items on the switch then it might be worth it, but just a TV is almost certainly pointless.