Connecting fresh-air damper to Lennox furnace OR replacing timer with new smart thermostat

We replaced a 20-year-old furnace with a new Lennox furnace and added air conditioning. The new system is still controlled by the 20-year-old Honeywell Chronotherm III thermostat. The furnace had (and still has) a fresh air intake with a Columbia fresh air damper controlled by a timer. The timer also turns on the whole-house exhaust fan, and USED to turn on the furnace fan. However, on the new Lennox furnace, the installer said the timer could not be used to turn on the furnace fan. So, we no longer have fresh air brought into the house based on the timer — I have to turn the timer on manually and turn the fan on via the old thermostat.

Can the timer be connected to the furnace fan without damage or risk? If so, how?

A far better option would be to replace the current thermostat and timer with a smart thermostat to control everything. However, ecobee support said:

Essentially, the damper and exhaust for the air intake is similar to a
ventilator if there is no timer involved. If it required some sort of
trigger from the HVAC unit in order to run, the thermostat will not be
able to detect which parts that are needed and will result in it not
engaging properly.

That's why there is no way for the damper to be connected directly to
the ecobee, as far as I know as the ecobee won't be able to
communicate with the timer or the damper .

Hypothetically, if there is no timer involved and the damper doesn't
require any trigger from the HVAC unit, it is possible to have it
connected to ecobee as an accessory (ventilator in this case). You can
view this article Troubleshooting accessory installations for more
info.

However, this is hypothetical as it all depends on the functionality
of the intake and how it responds to separate voltage. Since it's only
in theory, I would stick with my former answer that it's not possible
for the ecobee to control the damper.

Is it possible to use a new smart thermostat to control everything? (I prefer Honeywell or ecobee.) If so, how would I wire it to the Lennox furnace control board? (I assume I would remove the Honeywell timer.)

THANKS!