Three Houses and Engage started development at the same time with different development teams. That's partially why they're so different - the aim was for Engage to be different to Three Houses, the concept for which was nailed down in 2015 (but it was put on hold for Shadows of Valentia).
So the hypothetical timeline would go something like:
2015:
- Fates releases in Japan.
- Three Houses concept planning; put on hold.
- Shadows of Valentia enters development.
2017:
- Shadows of Valentia releases.
- Three Houses enters development, Intelligent Systems work with Koei Tecmo. This game is originally targeting 2018, believe it or not.
- Engage enters development with the goal being to provide a more traditional heroic fantasy game compared to Three Houses. This game targets 2020, the anniversary year for the series.
- Given Three Houses is Iron17 and Engage is Iron19, there may well have been plans for Iron18 at this point in time (Iron18 references appear in Engage's code). Perhaps it entered production for 3DS but was quickly cancelled; perhaps it never went beyond the planning stages and was canned entirely. If Iron18 is an FE4 remake, then perhaps Engage, as a new title, was prioritised over an FE4 remake given Three Houses was inspired by FE4 and so two titles of that vein in quick succession was deemed a bad idea. As we've seen already, Three Houses briefly began as a 3DS title in 2015 before being put on hold for 2 years, before becoming a Switch co-production with Koei-Tecmo. This should serve as a lesson that FE concepts can emerge, vanish, and re-emerge, and so the idea an FE4 remake was considered, put on hold, and then recommenced is something that has direct precedent.
2019:
- Three Houses releases after a delay.
2021:
- Engage is content complete by late 2021 after pandemic related delays. Staff might move to FE4 remake, which may or may not have been in the works for 3DS c. 2017 to 2019.
2023:
- Engage releases; FE4 is content complete.
2024:
- FE4 releases.
It's not as if Intelligent Systems do all this singlehandedly; there'll be outsourcing, some involvement from EPD and perhaps Koei-Tecmo are involved with the FE4 remake. The development timeline would certainly make sense; Three Houses DLC wrapped up in 2020, with the main game only having taken 2 years. It's also not unusual for Intelligent Systems to wrap up games well ahead of their release. WarioWare Get It Together went to age ratings agencies 12 months before it released, too. They're a pretty efficient developer.
Personally I'm not all that convinced an FE4 remake began on 3DS - interviews at the time Echoes released gave the impression it was the team's farewell to 3DS. However, given how accurate Emily and Nate were on Engage, given the existence of Iron18, and given IS have a clear track record balancing multiple projects and finishing titles ahead of release, I'm inclined to believe the FE4 remake is real and coming.