fix: fix bugs, improve type-stability

[AayushSabharwal]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[github-actions]

Benchmark Results (Julia vlts)

Time benchmarks

	master	5b34204...	master / 5b34204...
ODEProblem	6.88 ± 0.19 ms	6.72 ± 0.14 ms	1.02 ± 0.035
init	0.0642 ± 0.018 ms	0.0643 ± 0.018 ms	0.998 ± 0.39
large_parameter_init/ODEProblem	29.6 ± 2.1 ms	29.9 ± 2.1 ms	0.989 ± 0.098
large_parameter_init/init	0.0749 ± 0.018 ms	0.0733 ± 0.018 ms	1.02 ± 0.35
mtkcompile	9.84 ± 0.27 ms	9.74 ± 0.28 ms	1.01 ± 0.04
sparse_analytical_jacobian/ODEProblem	27 ± 1.6 ms	26.9 ± 1.2 ms	1 ± 0.076
sparse_analytical_jacobian/f_iip	0.062 ± 0.001 μs	0.063 ± 0 μs	0.984 ± 0.016
sparse_analytical_jacobian/f_oop	0.376 ± 0.012 ms	0.375 ± 0.011 ms	1 ± 0.044
time_to_load	5.65 ± 0.011 s	5.59 ± 0.089 s	1.01 ± 0.016

Memory benchmarks

	master	5b34204...	master / 5b34204...
ODEProblem	0.0317 M allocs: 1.94 MB	0.0317 M allocs: 1.94 MB	1
init	0.416 k allocs: 0.0697 MB	0.416 k allocs: 0.0697 MB	1
large_parameter_init/ODEProblem	0.317 M allocs: 11.5 MB	0.317 M allocs: 11.6 MB	1
large_parameter_init/init	0.604 k allocs: 0.172 MB	0.604 k allocs: 0.172 MB	1
mtkcompile	0.0602 M allocs: 3.46 MB	0.0602 M allocs: 3.46 MB	1
sparse_analytical_jacobian/ODEProblem	0.199 M allocs: 8.33 MB	0.199 M allocs: 8.33 MB	1
sparse_analytical_jacobian/f_iip	0 allocs: 0 B	0 allocs: 0 B
sparse_analytical_jacobian/f_oop	0.634 k allocs: 19.6 kB	0.634 k allocs: 19.6 kB	1
time_to_load	0.153 k allocs: 14.5 kB	0.153 k allocs: 14.5 kB	1

[github-actions]

Benchmark Results (Julia v1)

Time benchmarks

	master	5b34204...	master / 5b34204...
ODEProblem	8.7 ± 0.27 ms	6.76 ± 0.37 ms	1.29 ± 0.082
init	0.0548 ± 0.018 ms	0.0498 ± 0.018 ms	1.1 ± 0.52
large_parameter_init/ODEProblem	31.2 ± 4.9 ms	30.9 ± 5.9 ms	1.01 ± 0.25
large_parameter_init/init	0.0694 ± 0.034 ms	0.0676 ± 0.031 ms	1.03 ± 0.69
mtkcompile	9.76 ± 0.27 ms	8.83 ± 0.27 ms	1.11 ± 0.045
sparse_analytical_jacobian/ODEProblem	27.6 ± 3.3 ms	26.2 ± 3.4 ms	1.05 ± 0.19
sparse_analytical_jacobian/f_iip	0.09 ± 0.01 μs	0.09 ± 0.01 μs	1 ± 0.16
sparse_analytical_jacobian/f_oop	0.123 ± 0.013 ms	0.12 ± 0.013 ms	1.02 ± 0.16
time_to_load	5.66 ± 0.036 s	5.65 ± 0.011 s	1 ± 0.0067

Memory benchmarks

	master	5b34204...	master / 5b34204...
ODEProblem	0.0371 M allocs: 2.09 MB	31 k allocs: 1.82 MB	1.15
init	0.364 k allocs: 0.0453 MB	0.359 k allocs: 0.0458 MB	0.99
large_parameter_init/ODEProblem	0.329 M allocs: 12.5 MB	0.329 M allocs: 12.5 MB	1
large_parameter_init/init	0.798 k allocs: 0.151 MB	0.798 k allocs: 0.151 MB	1
mtkcompile	0.056 M allocs: 2.76 MB	0.0563 M allocs: 2.76 MB	0.998
sparse_analytical_jacobian/ODEProblem	0.189 M allocs: 7.28 MB	0.189 M allocs: 7.28 MB	1
sparse_analytical_jacobian/f_iip	0 allocs: 0 B	0 allocs: 0 B
sparse_analytical_jacobian/f_oop	0.848 k allocs: 27 kB	0.848 k allocs: 27 kB	1
time_to_load	0.145 k allocs: 11 kB	0.145 k allocs: 11 kB	1