Generating a 10MHz modulated signal
Solutions that come to mind:


Microcontroller + DAC: The use of DSP / DDS (direct digital synthesis) techniques on a fast microcontroller (preferably with an integrated FPU) to generate a frequency modulated sine wave could be a viable/inexpensive solution, with an internal or external (parallel) DAC, and an accurate external clock source. - A 10 MHz (carrier) and a 20 kHz modulation signal could probably be generated with a microcontroller from the STM32F/H series (by ST). - Lookup tables, ARM DSP optimizations and fix point math could help. - Whether the result satisfies your application's requirements depends on the necessary quality of the signal (sampling rate/depth, aliasing effects, jitter etc.). - The signal quality could probably be improved with an external analog filter.
FPGA + DAC: The signal could be generated with an FPGA and digital-to-analog converted with a fast parallel DAC, all clocked from an accurate clock source. The sine wave/carrier and the modulation signal (triangle) can be generated right inside the FPGA. (Instead of using a lookup table, the CORDIC way of implementing a sine function could come in handy). This could probably already be accomplished with a small/inexpensive FPGA from the ICE40 series (by Lattice).
Specialized IC: The direct digital synthesis (DDS) chips by Analog Devices might be a good choice, like you've mentioned above (?). (But I've never worked with them myself yet, so I can't tell...) - Compared to the other solutions above, specialized ICs are black boxes, and your solution will strongly depend on the supply of a single manufacturer.
Analog cirucit: The whole thing could be done with analog techniques, but that's usually harder, in my experience. And it is probably more expensive/difficult to get reliable/stable results.