That is correct. You cannot upgrade an original Avios booking using "Upgrade by Avios". You can only "upgrade by avios" cash bookings made via Aer Lingus.

Avios tickets are actually on BA ticket stock. Your ticket number will start with 125-, as opposed to EI's stock which starts in 053-. If you part payed with Avios on the EI website you will be issued a ticket on 053- stock, as these are revenue seats. It happens that the passenger chose to pay with Avios rather than Euro/Dollars/Sterling/Bitcoin/whatever.

If you have booked via avois.com, EI systems see the Avois ticket as a travel agent booking, so the EI agents can't do anything. In truth thats not dissimilar to BA's own Avios tickets which can only be changed by agents in the Executive Club. If there are reward Business class seats available you can pay a small change fee (around £35, from memory) and the difference in Avios. You could also cancel and rebook for the same fee. Hope this helps.

Long-time lurker, want to chime in as well on this:
Was successful upgrading a cash Y fare to J via avios on two occasions in the past year with EI (once very recently), out of two attempts. Both times were off-peak, requested a few months in advance of the flight and there was no avios reward availability in J on avios.com. First occasion was DUB>YYZ (J seatmap was at about 25% occupancy at the time) and second SFO>DUB (at around 40% occupancy). Booking became inaccessible via website/app on both occasions after upgrade was confirmed, and the only way to make changes or seat selection from then onwards was over the phone.

Email from EI acknowledging receipt of my request followed within 48-72 hours of me sending it, whereas the upgrade was confirmed around a week later.

A bit of a balancing act to be honest in terms of how long in advance to request the upgrade. Presumably, the earlier the better - to take advantage of lower J occupancy & higher chance of clearing. However be mindful that cash fare for J can be quite dynamic/unpredictable & discounted business fare buckets (e.g. "I") can go on sale - especially when J occupancy is low. Case in point:

- Booked SFO>DUB as part of a return leg, paid cash fare circa €1k for Economy Flex fare. At the time of booking, J cash fare was €3k+ even though J occupancy was low.
- Waited a couple of weeks & tracked J fares (with the idea of rebooking into J by paying the price difference, if J went on sale). J occupancy gradually crept up and I ultimately went ahead and asked for the Avios upgrade so as not to miss out altogether. 10 days after the avios upgrade was confirmed, two seats in "I" cash fare buckets in business were released at €1.3k :D.
- Needless to say, would have obviously preferred to pay the €300 fare difference to save 46k avios, get 125 TC instead of 75 TC and have a J ticket that is refundable & modifiable. (Whereas naturally avios upgrades, once confirmed, are non-refundable and non-transferrable if you end up having to rebook to a different flight/date).

I was looking for data points too before I made the request. Bought cash ticket for November returning day before thanksgiving from Ewr to Dublin. Ticket purchased Tuesday emailed Wednesday for Avios upgrade for two round trip. Upgrade awarded each way both travelers on Friday no additional cash fees total 148,000 Avios used. Easy process just took a couple of days.

Sorry for delay. One day to book outbound. One more day for return (I asked each separately).