Fix deprecated ODESystem and @mtkbuild usage in documentation examples

[ChrisRackauckas]

Summary

• Replace deprecated ODESystem with System constructor in README examples
• Replace deprecated @mtkbuild with @mtkcompile macro in demo.jl
• Update user-facing documentation to use current recommended syntax

Changes Made

• README.md: Fixed 3 instances of deprecated constructors and macros
• demo.jl: Fixed 6 instances of @mtkbuild → @mtkcompile

Motivation

These changes eliminate deprecation warnings that new users encounter when following documentation examples. The deprecation system properly redirects to new APIs, but updating examples prevents confusion.

Impact

• ✅ Better user onboarding experience
• ✅ Eliminates deprecation warnings in documentation
• ✅ Maintains same functionality with modern syntax
• ✅ No breaking changes (old APIs still work via deprecation layer)

Test plan

• Changes tested with existing test suite
• Documentation examples verified to work with new syntax
• Deprecation warnings eliminated

🤖 Generated with Claude Code

[github-actions]

Benchmark Results (Julia v1)

Time benchmarks

	master	5d03220...	master / 5d03220...
ODEProblem	0.0802 ± 0.0059 s	0.0808 ± 0.0055 s	0.993 ± 0.1
init	0.0574 ± 0.012 ms	0.0567 ± 0.012 ms	1.01 ± 0.31
large_parameter_init/ODEProblem	0.712 ± 0.015 s	0.703 ± 0.022 s	1.01 ± 0.038
large_parameter_init/init	0.1 ± 0.011 ms	0.1 ± 0.011 ms	0.999 ± 0.16
mtkcompile	0.0342 ± 0.00074 s	0.0341 ± 0.00045 s	1 ± 0.025
time_to_load	5.63 ± 0.027 s	5.58 ± 0.008 s	1.01 ± 0.005

Memory benchmarks

	master	5d03220...	master / 5d03220...
ODEProblem	0.578 M allocs: 18.8 MB	0.579 M allocs: 18.8 MB	0.999
init	0.815 k allocs: 0.0318 MB	0.815 k allocs: 0.0318 MB	1
large_parameter_init/ODEProblem	4.68 M allocs: 0.169 GB	4.68 M allocs: 0.169 GB	0.997
large_parameter_init/init	1.66 k allocs: 0.0712 MB	1.66 k allocs: 0.0712 MB	1
mtkcompile	0.232 M allocs: 8.08 MB	0.233 M allocs: 8.08 MB	0.999
time_to_load	0.159 k allocs: 11.2 kB	0.159 k allocs: 11.2 kB	1

[github-actions]

Benchmark Results (Julia vlts)

Time benchmarks

	master	5d03220...	master / 5d03220...
ODEProblem	0.089 ± 0.0067 s	0.0878 ± 0.0072 s	1.01 ± 0.11
init	0.062 ± 0.0023 ms	0.0618 ± 0.0021 ms	1 ± 0.05
large_parameter_init/ODEProblem	0.73 ± 0.0058 s	0.726 ± 0.0037 s	1.01 ± 0.0094
large_parameter_init/init	0.113 ± 0.0034 ms	0.113 ± 0.0031 ms	1 ± 0.041
mtkcompile	0.0374 ± 0.0021 s	0.0368 ± 0.0013 s	1.02 ± 0.067
time_to_load	5.91 ± 0.04 s	5.99 ± 0.044 s	0.986 ± 0.0098

Memory benchmarks

	master	5d03220...	master / 5d03220...
ODEProblem	0.588 M allocs: 20 MB	0.588 M allocs: 20 MB	1
init	0.908 k allocs: 0.0479 MB	0.908 k allocs: 0.0479 MB	1
large_parameter_init/ODEProblem	4.61 M allocs: 0.177 GB	4.61 M allocs: 0.177 GB	0.999
large_parameter_init/init	1.75 k allocs: 0.0874 MB	1.75 k allocs: 0.0874 MB	1
mtkcompile	0.233 M allocs: 8.64 MB	0.233 M allocs: 8.64 MB	1
time_to_load	0.153 k allocs: 14.5 kB	0.153 k allocs: 14.5 kB	1