Simplify availability checks - remove try/catch and symbol checking

Per review feedback, simplified the availability checks:
- Removed try/catch blocks as they're unnecessary
- Removed Libdl symbol checking - if the binary exists via is_available(), it's fine
- Just check if the JLL module is defined and is_available() returns true

This makes the code cleaner and follows the principle that if the binary
exists, we can trust it has the required symbols.

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <[email protected]>

Author: claude

src/mkl.jl

@@ -1,5 +1,3 @@
-using Libdl
-
 """
 ```julia
 MKLLUFactorization()
@@ -10,30 +8,12 @@ to avoid allocations and does not require libblastrampoline.
 """
 struct MKLLUFactorization <: AbstractFactorization end
 
-# Check if MKL is available and can be loaded
+# Check if MKL is available
 function __mkl_isavailable()
     if !@isdefined(MKL_jll)
         return false
     end
-    if !MKL_jll.is_available()
-        return false
-    end
-    # Try to load the library and check for required symbols
-    try
-        mkl_hdl = Libdl.dlopen(MKL_jll.libmkl_rt)
-        if mkl_hdl == C_NULL
-            return false
-        end
-        # Check for a required symbol
-        if Libdl.dlsym_e(mkl_hdl, "dgetrf_") == C_NULL
-            Libdl.dlclose(mkl_hdl)
-            return false
-        end
-        Libdl.dlclose(mkl_hdl)
-        return true
-    catch
-        return false
-    end
+    return MKL_jll.is_available()
 end
 
 function getrf!(A::AbstractMatrix{<:ComplexF64};

src/openblas.jl

@@ -1,5 +1,3 @@
-using Libdl
-
 """
 ```julia
 OpenBLASLUFactorization()
@@ -35,30 +33,12 @@ sol = solve(prob, OpenBLASLUFactorization())
 """
 struct OpenBLASLUFactorization <: AbstractFactorization end
 
-# Check if OpenBLAS is available and can be loaded
+# Check if OpenBLAS is available
 function __openblas_isavailable()
     if !@isdefined(OpenBLAS_jll)
         return false
     end
-    if !OpenBLAS_jll.is_available()
-        return false
-    end
-    # Try to load the library and check for required symbols
-    try
-        openblas_hdl = Libdl.dlopen(OpenBLAS_jll.libopenblas)
-        if openblas_hdl == C_NULL
-            return false
-        end
-        # Check for a required symbol
-        if Libdl.dlsym_e(openblas_hdl, "dgetrf_") == C_NULL
-            Libdl.dlclose(openblas_hdl)
-            return false
-        end
-        Libdl.dlclose(openblas_hdl)
-        return true
-    catch
-        return false
-    end
+    return OpenBLAS_jll.is_available()
 end
 
 function openblas_getrf!(A::AbstractMatrix{<:ComplexF64};