I honestly don't think it makes a difference. I left it until about 8 months with DS1 (might have done it once or twice before that) and he always took it fine. OTOH DS2 was mixed fed from birth, but had a period of rejecting it at about 4-5 months. But then later will happily drink water from a bottle teat, so he was probably just not wanting a bottle at that specific time - he used to have one at bedtime but always seemed to want breastfeeding soon afterwards anyway. I have known other children to just stop taking a bottle when they have been having one regularly, it's always about the same age (3-4 months ish).
Apparently this is because prior to this age drinking from a bottle is due to a sucking reflex. You put something of roughly the correct shape in their mouth and they will suck on it (you can notice this with your finger - older babies lick or bite, but don't tend to suck). Once they lose that reflex, then they have to re-learn how to do it and some babies take to this more easily than others. It seems to be a cultural belief in the UK that if you give a baby a bottle before this age then you'll override this but I don't see that it is necessarily the case. I think it's just confirmation bias when it works out that way.
Bottle preference/formula affecting supply isn't a myth but it is only a problem in a minority of situations, most people will be fine especially if you wait a few weeks after birth. If your baby is struggling to stimulate enough supply anyway due to being sleepy or having some kind of problem with latch then it's not a good idea to further reduce demand by filling them up with formula. Too much formula too early can stretch their tummies meaning it's harder to produce enough breastmilk to satisfy them (I had this problem with DS2). Also some babies come to prefer the faster flow of a bottle as it is less effort than feeding from the breast where they have to actually do some work. Likewise if they are struggling to latch for some reason, it's not a good idea to keep giving them bottles or dummies where the process of latching to them is quite different to a breast... that's just confusing the issue. And then lastly it's a more of a parent psychology thing rather than a direct effect on the baby - if you feel unconfident in your supply and keep giving the baby a bottle whenever you feel supply is low this has the opposite effect on your supply to what you want. It can also further increase feelings of insecurity about breastmilk as babies often settle more quickly after a bottle than they will after a breastfeed. If you think your supply is low you want to be offering the breast to the baby once an hour or more frequently to stimulate a lot more milk to be made.
If you do want to give a bottle early it's a good idea to stick to the slowest flow newborn sized teats and to do paced feeding of small amounts, as this helps prevent some of the reasons for bottle preference. Despite various promises there is no bottle which realistically mimics the motion of breastfeeding.
Whether you introduce a bottle or not I think it's really great when breastfeeding to bite the bullet and just get into leaving them, just using gaps between feeds. Obviously the gaps aren't always predictable, but starting with an hour or two at first stretching to about 4-6 hours once they are starting/established on solids. Even if they want to feed every half hour when you are around, a single 2 hour gap won't starve them and it gives a chance for other carers to develop different soothing and distraction methods which can be useful. They usually do absolutely fine and simply have a nice long feed as soon as you return.