I can't agree with that at all - it depends on the oil and the way they are stored, Grade 1 or whatever won't make any difference for some oils, citrus fruit oils in particular oxidise quite rapidly - a 1 yr shelf life is pretty normal. They are OK to use in burners, but on the skin they oxidise into terpenes, a skin irritant, doesn't matter how good the oil is, the essential oil molecules readily oxidise. Some essential oils can last for thousands of years, because they don't oxidise- the resin s such as frankincense and myrrhare good examples. most of the wood essential oils havefairly long shelf lives too, afterall the original treeslast for years, but fruits and flowery oils often have relatively shorter shelf lives, citrus oils being about the shortest.
If you store your oils incorrectly- clear plastic container in the sun, nothing is going to help it, regardless of the quality of the oil, which is why essential oils are generally available in dark glass bottles. If you keep your oils in a hot environment they will breakdown faster too.
If the chemical compounds in the oil are fairly reactive, eg they oxidise quickly, the quality of the oil makes little difference past its use by date.
Where the oils come from, the soil conditions, how the essential oils were extracted, etc, all affect the chemical compounds in the essential oils, eg ginger steam distilled vs ginger CO2 extract may as well be 2 different oils.
The problem you have with different suppliers, is you don't know how long the essential oil was sitting in some vat on the farm before it went to a middleman that kept it stored in the sun until the prices went up enough to make it worthwhile selling, then how long the wholesaler had the oil, and if you're buying retail, how long it has been sitting on the shelf in the shop. Reputable suppliers will know harvest dates of the batch and apply the use by date applicable. If your oils don't have use by dates, I'd be cautious about the brand in general, and I would be very cautious about citrus fruit oils.