Nyxbloom Ancient replaces only the quantity of mana generated, and nothing else. You'll still only be able to cast Eldrazi spells off of Eldrazi Temple, and you'll still take damage from your Sulfurous Springs. So, yes, you will still be able to place a counter on your Autogenerator.
The release notes for Theros Beyond Death state:
Nyxbloom Ancient doesn't produce any mana itself. Rather, it causes permanents you tap for mana to produce more mana. If the mana ability of that permanent puts any restrictions or riders on the mana it produces, that will apply to all the mana it produces this way.
Some additional support from the rules:
614.1a Effects that use the word “instead” are replacement effects. Most replacement effects use the word “instead” to indicate what events will be replaced with other events.
Nyxbloom Ancient says "it produces three times as much of that mana instead". This indicates that only the quantity of mana produced is being replaced.
614.11b If an effect would have a player both draw a card and perform an additional action on that card, and the draw is replaced, the additional action is not performed on any cards that are drawn as a result of that replacement effect.
This rule specifically calls out that replacing a card draw + rider with an effect also overrules the rider. There is no similar rule for mana abilities, which implies that the riders still take effect.