Aircraft radomes?

Some airborne radar may have IFF interrogator antenna elements applied to an *antenna* radome, which itself may be tucked behind the aircraft radome.

An aircraft nose radome may have embedded lightning diverter strips. That would normally preclude also embedding antenna elements into the same space, due to interactions impacting the antenna performance, unless the two designs were merged into one (grounded antenna concepts). Or they could use those 'stepping stone' diverters that are RF transparent.

"...compromises in antenna size will reduce antenna gain..."

Many airborne antenna applications do not want high antenna gain, because antenna gain means directionality and the aircraft may be in too many variable orientations for significant antenna gain to be useful. Often the system designer wants isotropic coverage approaching 0dBi gain and complete spherical coverage. This doesn't apply to moveable antennas such as a high gain steerable antenna system.

All above is perfectly public info.

I'm really unclear on what you think the problem is. Most military aircraft use arrayed antennas behind the domes you refer to. The big issue is that the signal phase has to be extremely well controlled, and that can only be accomplished with flat panels and precisely placed receiver elements.

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I think that the proposal is to take advantage of (for example) the nose radome, by embedding antenna array elements directly into it.

This concept could easily have a niche application somewhere. I wouldn't be surprised if somebody's fibreglass tail cap might have an embedded VHF element.

With computer tools, a phased array design could conceivably be wrapped around any (reasonable) arbitrary surface, and then optimized. The element phasing would have to individually adjusted. And then the resultant performance may not meet the system requirements.

That antenna in the picture is inside the nose radome, so putting another antenna on the outside nullifies all that hardware on the inside. The radome also serves to protect the antenna from damage, like from hail and rain. Having the antenna external isn't necessarily an optimal solution.

I think consideration of the full field of regard of the antenna, and the gain requirements would show that this design is/was optimal for its cost/benefit.

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"...this design is/was optimal..."

For the particular example represented by the picture that you've selected (above), of course that design is almost certainly optimal. For example, it would be rather awkward to have a high gain airborne radar antenna embedded into a fixed-aim aircraft nose radome where the entire aircraft would have to be steered to scan the radar beam. !!

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The counter example where an antenna array embedded into a radome might actually be feasible would be the example of VHF/UHF Homing, where an antenna array (often of just two elements) is used to Home in on (this is distinct from DF-ing) signals, e.g. ELTs for Search and Rescue (SAR). In that case, having to aim the entire SAR aircraft towards the desired signal is standard operating procedure for Homing. Because they need to fly over the target. So a nose radome with embedded VHF/UHF elements for Homing might be useful.

The overall radome design would have to include consideration of the lightning diverter strips (interactions) as previously mentioned. Also, P-static would be another point to carefully consider. Finally there are factors about actual costs and maintenance issues to be traded off; it would be easy to arrive at a solution that's actually worse than just using normal antennas.