Yes the non isolated chassis was common till the age of the VCR, because the TV set only had an antenna input, and a headphone jack, both of which are easy to isolate, using 2 1n 3kV capacitors for the antenna input, and a simple robust insulation audio transformer to drive the speaker, also having as bonus matching for the 16R speaker to the 3R output impedance of the transistor output stage, running off around 16V. Non isolated just meant plastic shafts for all the controls, and long shafts as well, to place them at least 5mm inside from the plastic or wood case and front, and a plastic rear cover with a warning label, and a detacheable mains cable that came off with the rear backing.
Then came the first generation of VCR's, and a demand arose for a way to connect audio and video into the set, helped by the adoption of SCART in the European area, which predates the EU, but where the assorted countries wanted to harmonise on TV standards, and have a common interconnection for all this consumer audio and video gear. So the first ones used a non isolated chassis, and had a small power supply that provided an isolated supply for this circuitry, with assorted forms of transfer across the isolation barrier, from photodiodes and phototransistors, to modulating it on a 20MHz or so carrier and using some 10pF capacitors to carry it, to using RF transformers to do the same. Then you got sets later on that had a isolated control and input, and non isolated HV generation, deflection and such, like Salora, who used the LOPT as both power supply and isolation, with the primary drive and horizontal deflection running non isolated, but EHT, 200V CRT drive, heater and all other parts running off isolated windings on the LOPT, and then a low power SMPS doing the standby supply for the microcontroller and the jungle chip, which performed all the set functions in a single package, all controlled over I2C to set up hundreds of internal registers to determine bands covered, IF bandwidths, colour carriers and TV system along with audio types, all in one single package.
Till the final versions isolated all the TV functions, using a SMPS to provide a 5V supply for the microcontroller, and a second switchable one that provided all the other rails when commanded, so the TV set would be in standby till the SCART input commanded it on, or video was present, or the IR remote powered it on.
As to asymmetric loading, the first generation Phillips G11 used a thyristor power supply, and, as the UK has polarised sockets and plugs, and these came with a 2 wire lead with factory attached plug, they all were wired identically, so they were responsible for a good number of older distribution transformers catching fire, as the 11.6A pulse of current every second mains cycle all added up, in residential areas, to a massive DC current flow in the older transformers, which would then have the core saturate, and then blow fuses, or overheat and catch fire from massive primary side current as the inductance dropped to basically air cored. Rewinds put in a distributed oil gap instead, stacking the laminations to give a small gap between them in the centre of the core, instead of a butt join, and a slight correction to the lamination stack and windings to compensate, plus bigger oil coolers.